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126-472: (Redirected from Two States ) Two-state (or variants) may refer to: Two-state quantum system , in physics Two-state trajectory in biophysics Two-state solution , a proposed solution to the Israeli-Palestinian conflict Two-state solution (Cyprus) , a proposed permanent division of the island of Cyprus 2 States: The Story of My Marriage ,

252-486: A {\displaystyle a} and b {\displaystyle b} . Such a state vector corresponds to a Bloch vector in the xz -plane making an angle tan ⁡ ( θ / 2 ) = b / a {\displaystyle \tan(\theta /2)=b/a} with the z -axis. This vector will proceed to precess around z ^ {\displaystyle \mathbf {\hat {z}} } . In theory, by allowing

378-1412: A Pauli matrix σ i {\displaystyle \sigma _{i}} and the conjugate transpose of the wavefunction, and subsequently expanding the product of two Pauli matrices yields ψ † σ j ∂ ψ ∂ t = i μ ℏ ψ † σ j σ i B i ψ = i μ ℏ ψ † ( I δ i j − i σ k ε i j k ) B i ψ = μ ℏ ψ † ( i I δ i j + σ k ε i j k ) B i ψ {\displaystyle \psi ^{\dagger }\sigma _{j}{\frac {\partial \psi }{\partial t}}=i{\frac {\mu }{\hbar }}\psi ^{\dagger }\sigma _{j}\sigma _{i}B_{i}\psi =i{\frac {\mu }{\hbar }}\psi ^{\dagger }\left(I\delta _{ij}-i\sigma _{k}\varepsilon _{ijk}\right)B_{i}\psi ={\frac {\mu }{\hbar }}\psi ^{\dagger }\left(iI\delta _{ij}+\sigma _{k}\varepsilon _{ijk}\right)B_{i}\psi } Adding this equation to its own conjugate transpose yields

504-1053: A 2×2 hermitian matrix , H = ( ⟨ 1 | H | 1 ⟩ ⟨ 1 | H | 2 ⟩ ⟨ 2 | H | 1 ⟩ ⟨ 2 | H | 2 ⟩ ) = ( H 11 H 12 H 12 ∗ H 22 ) . {\displaystyle \mathbf {H} ={\begin{pmatrix}\langle 1|H|1\rangle &\langle 1|H|2\rangle \\\langle 2|H|1\rangle &\langle 2|H|2\rangle \end{pmatrix}}={\begin{pmatrix}H_{11}&H_{12}\\H_{12}^{*}&H_{22}\end{pmatrix}}.} The time-independent Schrödinger equation states that H | ψ ⟩ = E | ψ ⟩ {\displaystyle H|\psi \rangle =E|\psi \rangle } ; substituting for | ψ ⟩ {\displaystyle |\psi \rangle } in terms of

630-591: A body for frequency ν at absolute temperature T is given by where k B {\displaystyle k_{\text{B}}} is the Boltzmann constant , h {\displaystyle h} is the Planck constant, and c {\displaystyle c} is the speed of light in the medium, whether material or vacuum. The spectral radiance of a body, B ν {\displaystyle B_{\nu }} , describes

756-792: A collection of spin-1/2 particles can be derived from the time dependent Schrödinger equation for a two level system. Starting with the previously stated Hamiltonian i ℏ ∂ t ψ = − μ σ ⋅ B ψ {\displaystyle i\hbar \partial _{t}\psi =-\mu {\boldsymbol {\sigma }}\cdot \mathbf {B} \psi } , it can be written in summation notation after some rearrangement as ∂ ψ ∂ t = i μ ℏ σ i B i ψ {\displaystyle {\frac {\partial \psi }{\partial t}}=i{\frac {\mu }{\hbar }}\sigma _{i}B_{i}\psi } Multiplying by

882-487: A complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit . Two-state systems are the simplest quantum systems that are of interest, since the dynamics of a one-state system is trivial (as there are no other states in which the system can exist). The mathematical framework required for the analysis of two-state systems is that of linear differential equations and linear algebra of two-dimensional spaces. As

1008-449: A description of absorption or decay, because such processes require coupling to a continuum. Such processes would involve exponential decay of the amplitudes, but the solutions of the two-state system are oscillatory. Supposing the two available basis states of the system are | 1 ⟩ {\displaystyle |1\rangle } and | 2 ⟩ {\displaystyle |2\rangle } , in general

1134-409: A domestic lightbulb; that is to say that the spotlight gives out more energy per unit time and per unit space (and hence consumes more electricity) than the ordinary bulb, even though the color of the light might be very similar. Other waves, such as sound or the waves crashing against a seafront, also have their intensity. However, the energy account of the photoelectric effect did not seem to agree with

1260-777: A general state.) This results in a diagonal matrix with the diagonal elements being the energies of the eigenstates and the off-diagonal elements being zero. The form of the matrix above that uses bra-ket-enclosed Hamiltonians is a more generalized version of this matrix. One might ask why it is necessary to write the Hamiltonian matrix in such a general form with bra-ket-enclosed Hamiltonians, since H i j , i ≠ j {\displaystyle H_{ij},i\neq j} should always equal zero and H i i {\displaystyle H_{ii}} should always equal ε i {\displaystyle \varepsilon _{i}} . The reason

1386-665: A left hand side of the form ψ † σ j ∂ ψ ∂ t + ∂ ψ † ∂ t σ j ψ = ∂ ( ψ † σ j ψ ) ∂ t {\displaystyle \psi ^{\dagger }\sigma _{j}{\frac {\partial \psi }{\partial t}}+{\frac {\partial \psi ^{\dagger }}{\partial t}}\sigma _{j}\psi ={\frac {\partial \left(\psi ^{\dagger }\sigma _{j}\psi \right)}{\partial t}}} And

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1512-468: A left-handed manner. In general, by a rotation around z ^ {\displaystyle \mathbf {\hat {z}} } , any state vector ψ ( 0 ) {\displaystyle \psi (0)} can be represented as a | ↑ ⟩ + b | ↓ ⟩ {\displaystyle a\left|\uparrow \right\rangle +b\left|\downarrow \right\rangle } with real coefficients

1638-423: A mathematical expression that could reproduce Wien's law (for short wavelengths) and the empirical formula (for long wavelengths). This expression included a constant, h {\displaystyle h} , which is thought to be for Hilfsgrösse (auxiliary variable), and subsequently became known as the Planck constant. The expression formulated by Planck showed that the spectral radiance per unit frequency of

1764-494: A novel by Chetan Bhagat 2 States (2014 film) , a Bollywood film based on the novel 2 States (soundtrack) , soundtrack by Shankar–Ehsaan–Loy for the film 2 States (2020 film) , an Indian Malayalam-language film Two Chinas Special state-to-state relations , also called "Two-state theory", proposed by Lee Teng-hui in 1999. See also [ edit ] Two Nations theory (disambiguation) Two-state solution (disambiguation) Topics referred to by

1890-530: A pair of coupled linear equations, but this time they are first order partial differential equations: i ℏ ∂ t c = H c {\textstyle i\hbar \partial _{t}\mathbf {c} =\mathbf {Hc} } . If H {\displaystyle \mathbf {H} } is time independent there are several approaches to find the time dependence of c 1 , c 2 {\displaystyle c_{1},c_{2}} , such as normal modes . The result

2016-406: A photon with angular frequency ω = 2 πf is given by while its linear momentum relates to where k is an angular wavenumber . These two relations are the temporal and spatial parts of the special relativistic expression using 4-vectors . Classical statistical mechanics requires the existence of h (but does not define its value). Eventually, following upon Planck's discovery, it

2142-467: A plurality of photons, whose energetic sum is greater than the energy of the photoelectrons, acts virtually simultaneously (multiphoton effect). Assuming the frequency is high enough to cause the photoelectric effect, a rise in intensity of the light source causes more photoelectrons to be emitted with the same kinetic energy, rather than the same number of photoelectrons to be emitted with higher kinetic energy. Einstein's explanation for these observations

2268-963: A restrictive condition on a general state vector to yield an eigenvector of H {\displaystyle H} , exactly analogous to the time-independent Schrödinger equation. Of course, in general, commuting the matrix with a state vector will not result in the same vector multiplied by a constant E. For general validity, one has to write the equation in the form ( H 11 H 12 H 12 ∗ H 22 ) ( c 1 c 2 ) = ( ε 1 c 1 ε 2 c 2 ) , {\displaystyle {\begin{pmatrix}H_{11}&H_{12}\\H_{12}^{*}&H_{22}\end{pmatrix}}{\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}}={\begin{pmatrix}\varepsilon _{1}c_{1}\\\varepsilon _{2}c_{2}\end{pmatrix}},} with

2394-446: A result, the dynamics of a two-state system can be solved analytically without any approximation. The generic behavior of the system is that the wavefunction's amplitude oscillates between the two states. A well known example of a two-state system is the spin of a spin-1/2 particle such as an electron, whose spin can have values + ħ /2 or − ħ /2, where ħ is the reduced Planck constant . The two-state system cannot be used as

2520-1060: A right hand side of the form μ ℏ ψ † ( i I δ i j + σ k ε i j k ) B i ψ + μ ℏ ψ † ( − i I δ i j + σ k ε i j k ) B i ψ = 2 μ ℏ ( ψ † σ k ψ ) B i ε i j k {\displaystyle {\frac {\mu }{\hbar }}\psi ^{\dagger }\left(iI\delta _{ij}+\sigma _{k}\varepsilon _{ijk}\right)B_{i}\psi +{\frac {\mu }{\hbar }}\psi ^{\dagger }\left(-iI\delta _{ij}+\sigma _{k}\varepsilon _{ijk}\right)B_{i}\psi ={\frac {2\mu }{\hbar }}\left(\psi ^{\dagger }\sigma _{k}\psi \right)B_{i}\varepsilon _{ijk}} As previously mentioned,

2646-782: A state vector ψ ( 0 ) {\displaystyle \psi (0)} that is a normalized superposition of | ↑ ⟩ {\displaystyle \left|\uparrow \right\rangle } and | ↓ ⟩ {\displaystyle \left|\downarrow \right\rangle } , that is, a vector that can be represented in the σ z {\displaystyle \sigma _{z}} basis as ψ ( 0 ) = 1 2 ( 1 1 ) {\displaystyle \psi (0)={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\1\end{pmatrix}}} The components of ψ ( t ) {\displaystyle \psi (t)} on

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2772-447: A system of two linear equations that can be combined into the above matrix equation. Like before, this can only be satisfied if c 1 {\displaystyle c_{1}} or c 2 {\displaystyle c_{2}} is zero, and when this happens, the constant E {\displaystyle E} will be the energy of the remaining state. The above matrix equation should thus be interpreted as

2898-405: A time evolution operator acting on a general spin state of a spin-1/2 particle will lead to the precession about the axis defined by the applied magnetic field (this is the quantum mechanical equivalent of Larmor precession ) The above method can be applied to the analysis of any generic two-state system that is interacting with some field (equivalent to the magnetic field in the previous case) if

3024-499: A transverse rf field B 1 rotating in the xy -plane in a right-handed fashion around B 0 : B = ( B 1 cos ⁡ ω r t B 1 sin ⁡ ω r t B 0 ) . {\displaystyle \mathbf {B} ={\begin{pmatrix}B_{1}\cos \omega _{\mathrm {r} }t\\B_{1}\sin \omega _{\mathrm {r} }t\\B_{0}\end{pmatrix}}.} As in

3150-897: A two-dimensional complex Hilbert space . Thus the state vector corresponding to the state | ψ ⟩ {\displaystyle |\psi \rangle } is | ψ ⟩ ≡ ( ⟨ 1 | ψ ⟩ ⟨ 2 | ψ ⟩ ) = ( c 1 c 2 ) = c 1 ( 1 0 ) + c 2 ( 0 1 ) = c , {\displaystyle |\psi \rangle \equiv {\begin{pmatrix}\langle 1|\psi \rangle \\\langle 2|\psi \rangle \end{pmatrix}}={\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}}=c_{1}{\begin{pmatrix}1\\0\end{pmatrix}}+c_{2}{\begin{pmatrix}0\\1\end{pmatrix}}=\mathbf {c} ,} and

3276-432: A two-state system as long as the observable one is interested in has two eigenvalues. For example, a spin-1/2 particle may in reality have additional translational or even rotational degrees of freedom, but those degrees of freedom are irrelevant to the preceding analysis. Mathematically, the neglected degrees of freedom correspond to the degeneracy of the spin eigenvalues. Another case where the effective two-state formalism

3402-1656: Is P i ( t ) = | c i ( t ) | 2 = | U i 1 ( t ) | 2 {\displaystyle P_{i}(t)=|c_{i}(t)|^{2}=|U_{i1}(t)|^{2}} . In the case of the starting state, P 1 ( t ) = | c 1 ( t ) | 2 = | U 11 ( t ) | 2 {\displaystyle P_{1}(t)=|c_{1}(t)|^{2}=|U_{11}(t)|^{2}} , and from above, U 11 ( t ) = e − i α t ℏ ( cos ⁡ ( | r | ℏ t ) − i sin ⁡ ( | r | ℏ t ) δ | r | ) . {\displaystyle U_{11}(t)=e^{\frac {-i\alpha t}{\hbar }}\left(\cos \left({\frac {|\mathbf {r} |}{\hbar }}t\right)-i\sin \left({\frac {|\mathbf {r} |}{\hbar }}t\right){\frac {\delta }{|\mathbf {r} |}}\right).} Hence, P 1 ( t ) = cos 2 ⁡ ( Ω t ) + sin 2 ⁡ ( Ω t ) Δ 2 Ω 2 . {\displaystyle P_{1}(t)=\cos ^{2}(\Omega t)+\sin ^{2}(\Omega t){\frac {\Delta ^{2}}{\Omega ^{2}}}.} Obviously, P 1 ( 0 ) = 1 {\displaystyle P_{1}(0)=1} due to

3528-491: Is W · sr · m · Hz , while that of B λ {\displaystyle B_{\lambda }} is W·sr ·m . Planck soon realized that his solution was not unique. There were several different solutions, each of which gave a different value for the entropy of the oscillators. To save his theory, Planck resorted to using the then-controversial theory of statistical mechanics , which he described as "an act of desperation". One of his new boundary conditions

3654-569: Is Rabi flopping from guaranteed occupation of state 1, to guaranteed occupation of state 2, and back to state 1, etc., with frequency | Ω R | {\displaystyle |\Omega _{R}|} . As the detuning is increased away from zero, the frequency of the flopping increases (to Ω ) and the amplitude of exciting the electron decreases to Ω 2 / Δ 2 {\displaystyle \Omega ^{2}/\Delta ^{2}} . For time dependent Hamiltonians induced by light waves, see

3780-1165: Is a 2×2 matrix eigenvalues and eigenvectors problem. As mentioned above, this equation comes from plugging a general state into the time-independent Schrödinger equation. Remember that the time-independent Schrödinger equation is a restrictive condition used to specify the eigenstates. Therefore, when plugging a general state into it, we are seeing what form the general state must take to be an eigenstate. Doing so, and distributing, we get c 1 H | 1 ⟩ + c 2 H | 2 ⟩ = c 1 E | 1 ⟩ + c 2 E | 2 ⟩ {\displaystyle c_{1}H|1\rangle +c_{2}H|2\rangle =c_{1}E|1\rangle +c_{2}E|2\rangle } , which requires c 1 {\displaystyle c_{1}} or c 2 {\displaystyle c_{2}} to be zero ( E {\displaystyle E} cannot be equal to both ε 1 {\displaystyle \varepsilon _{1}} and ε 2 {\displaystyle \varepsilon _{2}} ,

3906-529: Is also common to refer to this ℏ {\textstyle \hbar } as "Planck's constant" while retaining the relationship ℏ = h / ( 2 π ) {\textstyle \hbar =h/(2\pi )} . By far the most common symbol for the reduced Planck constant is ℏ {\textstyle \hbar } . However, there are some sources that denote it by h {\textstyle h} instead, in which case they usually refer to it as

Two-state - Misplaced Pages Continue

4032-636: Is an experimentally determined constant (the Rydberg constant ) and n ∈ { 1 , 2 , 3 , . . . } {\displaystyle n\in \{1,2,3,...\}} . This approach also allowed Bohr to account for the Rydberg formula , an empirical description of the atomic spectrum of hydrogen, and to account for the value of the Rydberg constant R ∞ {\displaystyle R_{\infty }} in terms of other fundamental constants. In discussing angular momentum of

4158-673: Is called the generalised Rabi frequency, Ω R = ( β + i γ ) / ℏ {\displaystyle \Omega _{R}=(\beta +i\gamma )/\hbar } is called the Rabi frequency, and Δ = δ / ℏ {\displaystyle \Delta =\delta /\hbar } is called the detuning. At zero detuning, P 1 ( t ) = cos 2 ⁡ ( | Ω R | t ) {\displaystyle P_{1}(t)=\cos ^{2}(|\Omega _{R}|t)} , i.e., there

4284-429: Is different from Wikidata All article disambiguation pages All disambiguation pages Two-state quantum system In quantum mechanics , a two-state system (also known as a two-level system ) is a quantum system that can exist in any quantum superposition of two independent (physically distinguishable) quantum states . The Hilbert space describing such a system is two- dimensional . Therefore,

4410-1237: Is easily proved that U ( t ) {\displaystyle \mathbf {U} (t)} is unitary , meaning that U † U = 1 {\displaystyle \mathbf {U} ^{\dagger }\mathbf {U} =1} . It can be shown that U ( t ) = e − i H t / ℏ = e − i α t / ℏ ( cos ⁡ ( | r | ℏ t ) σ 0 − i sin ⁡ ( | r | ℏ t ) r ^ ⋅ σ ) , {\displaystyle \mathbf {U} (t)=e^{-i\mathbf {H} t/\hbar }=e^{-i\alpha t/\hbar }\left(\cos \left({\frac {|\mathbf {r} |}{\hbar }}t\right)\sigma _{0}-i\sin \left({\frac {|\mathbf {r} |}{\hbar }}t\right){\hat {r}}\cdot {\boldsymbol {\sigma }}\right),} where r ^ = r | r | . {\textstyle {\hat {r}}={\frac {\mathbf {r} }{|\mathbf {r} |}}.} When one changes

4536-1532: Is easily seen to be given by: U ( t ) = e − i H t / ℏ = ( e − i E 1 t / ℏ 0 0 e − i E 2 v t / ℏ ) = e − i α t / ℏ ( e − i δ t / ℏ 0 0 e i δ t / ℏ ) = e − i α t / ℏ ( cos ⁡ ( δ ℏ t ) σ 0 − i sin ⁡ ( δ ℏ t ) σ 3 ) . {\displaystyle \mathbf {U} (t)=e^{-i\mathbf {H} t/\hbar }={\begin{pmatrix}e^{-iE_{1}t/\hbar }&0\\0&e^{-iE_{2}vt/\hbar }\end{pmatrix}}=e^{-i\alpha t/\hbar }{\begin{pmatrix}e^{-i\delta t/\hbar }&0\\0&e^{i\delta t/\hbar }\end{pmatrix}}=e^{-i\alpha t/\hbar }\left(\cos \left({\frac {\delta }{\hbar }}t\right)\sigma _{0}-i\sin \left({\frac {\delta }{\hbar }}t\right){\boldsymbol {\sigma }}_{3}\right).} The e − i α t / ℏ {\displaystyle e^{-i\alpha t/\hbar }} factor merely contributes to

4662-510: Is expressed with the unit joule per hertz (J⋅Hz ) or joule-second (J⋅s). The above values have been adopted as fixed in the 2019 revision of the SI . Since 2019, the numerical value of the Planck constant has been fixed, with a finite decimal representation. This fixed value is used to define the SI unit of mass, the kilogram : "the kilogram [...] is defined by taking the fixed numerical value of h to be 6.626 070 15 × 10 when expressed in

4788-478: Is extremely small in terms of ordinarily perceived everyday objects. Since the frequency f , wavelength λ , and speed of light c are related by f = c λ {\displaystyle f={\frac {c}{\lambda }}} , the relation can also be expressed as In 1923, Louis de Broglie generalized the Planck–Einstein relation by postulating that the Planck constant represents

4914-458: Is given as e i ω t σ ⋅ n ^ = ( e i ω t 0 0 e − i ω t ) . {\displaystyle e^{i\omega t{\boldsymbol {\sigma }}\cdot \mathbf {\hat {n}} }={\begin{pmatrix}e^{i\omega t}&0\\0&e^{-i\omega t}\end{pmatrix}}.} It can be seen that such

5040-602: Is given by H = ( ε 1 β − i γ β + i γ ε 2 ) , {\displaystyle \mathbf {H} ={\begin{pmatrix}\varepsilon _{1}&\beta -i\gamma \\\beta +i\gamma &\varepsilon _{2}\end{pmatrix}},} where ε 1 , ε 2 , β {\displaystyle \varepsilon _{1},\varepsilon _{2},\beta } and γ are real numbers with units of energy. The allowed energy levels of

5166-417: Is given by ( β , γ , δ ) {\displaystyle (\beta ,\gamma ,\delta )} and σ {\displaystyle \sigma } is given by ( σ 1 , σ 2 , σ 3 ) {\displaystyle (\sigma _{1},\sigma _{2},\sigma _{3})} . This representation simplifies

Two-state - Misplaced Pages Continue

5292-546: Is known by many other names: reduced Planck's constant ), the rationalized Planck constant (or rationalized Planck's constant , the Dirac constant (or Dirac's constant ), the Dirac h {\textstyle h} (or Dirac's h {\textstyle h} ), the Dirac ℏ {\textstyle \hbar } (or Dirac's ℏ {\textstyle \hbar } ), and h-bar . It

5418-475: Is made of the rotating wave approximation in deriving such results. Here the Schrödinger equation reads − μ σ ⋅ B ψ = i ℏ ∂ ψ ∂ t . {\displaystyle -\mu {\boldsymbol {\sigma }}\cdot \mathbf {B} \psi =i\hbar {\frac {\partial \psi }{\partial t}}.} Expanding

5544-401: Is one of the smallest constants used in physics. This reflects the fact that on a scale adapted to humans, where energies are typical of the order of kilojoules and times are typical of the order of seconds or minutes, the Planck constant is very small. When the product of energy and time for a physical event approaches the Planck constant, quantum effects dominate. Equivalently, the order of

5670-433: Is only satisfied by eigenstates of H, which are (by definition of the state vector) the states where all except one coefficient are zero. Now, if we follow the same derivation, but before acting with the Hamiltonian on the individual states, we multiply both sides by ⟨ 1 | {\displaystyle \langle 1|} or ⟨ 2 | {\displaystyle \langle 2|} , we get

5796-642: Is sufficiently strong, some proportion of the spins will be pointing directly down prior to the introduction of the rotating field. If the angular frequency of the rotating magnetic field is chosen such that ω r = − 2 ω 0 {\displaystyle \omega _{r}=-2\omega _{0}} , in the rotating frame the state vector will precess around x ^ {\displaystyle {\hat {x}}} with frequency 2 ω 1 {\displaystyle 2\omega _{1}} , and will thus flip from down to up releasing energy in

5922-432: Is that c ( t ) = e − i H t / ℏ c 0 = U ( t ) c 0 . {\displaystyle \mathbf {c} (t)=e^{-i\mathbf {H} t/\hbar }\mathbf {c} _{0}=\mathbf {U} (t)\mathbf {c} _{0}.} where c 0 = c ( 0 ) {\displaystyle \mathbf {c} _{0}=\mathbf {c} (0)}

6048-427: Is that, in some more complex problems, the state vectors may not be eigenstates of the Hamiltonian used in the matrix. One place where this occurs is in degenerate perturbation theory , where the off-diagonal elements are nonzero until the problem is solved by diagonalization . Because of the hermiticity of H {\displaystyle \mathbf {H} } the eigenvalues are real; or, rather, conversely, it

6174-477: Is the gyromagnetic ratio γ {\displaystyle \gamma } , yields another form for the equations of motion of the Bloch vector ∂ R j ∂ t = γ R k B i ε k i j {\displaystyle {\frac {\partial R_{j}}{\partial t}}=\gamma R_{k}B_{i}\varepsilon _{kij}} where

6300-935: Is the 2×2 identity matrix and the matrices σ k {\displaystyle \sigma _{k}} with k = 1 , 2 , 3 {\displaystyle k=1,2,3} are the Pauli matrices . This decomposition simplifies the analysis of the system, especially in the time-independent case, where the values of α , β , γ {\displaystyle \alpha ,\beta ,\gamma } and δ {\displaystyle \delta } are constants. The Hamiltonian can be further condensed as H = α ⋅ σ 0 + r ⋅ σ . {\displaystyle \mathbf {H} =\alpha \cdot \sigma _{0}+\mathbf {r} \cdot {\boldsymbol {\sigma }}.} The vector r {\displaystyle \mathbf {r} }

6426-440: Is the average energy of the two levels, and the norm of r {\displaystyle \mathbf {r} } is the splitting between them. The corresponding eigenvectors are denoted as | + ⟩ {\displaystyle |+\rangle } and | − ⟩ {\displaystyle |-\rangle } . We now assume that the probability amplitudes are time-dependent, though

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6552-411: Is the emission of electrons (called "photoelectrons") from a surface when light is shone on it. It was first observed by Alexandre Edmond Becquerel in 1839, although credit is usually reserved for Heinrich Hertz , who published the first thorough investigation in 1887. Another particularly thorough investigation was published by Philipp Lenard (Lénárd Fülöp) in 1902. Einstein's 1905 paper discussing

6678-1328: Is the magnitude of the particle's magnetic moment and σ {\displaystyle {\boldsymbol {\sigma }}} is the vector of Pauli matrices . Solving the time dependent Schrödinger equation H ψ = i ℏ ∂ t ψ {\displaystyle H\psi =i\hbar \partial _{t}\psi } yields ψ ( t ) = e i ω t σ ⋅ n ^ ψ ( 0 ) , {\displaystyle \psi (t)=e^{i\omega t{\boldsymbol {\sigma }}\cdot \mathbf {\hat {n}} }\psi (0),} where ω = μ B / ℏ {\displaystyle \omega =\mu B/\hbar } and e i ω t σ ⋅ n ^ = cos ⁡ ( ω t ) I + i n ^ ⋅ σ sin ⁡ ( ω t ) {\displaystyle e^{i\omega t{\boldsymbol {\sigma }}\cdot \mathbf {\hat {n}} }=\cos {\left(\omega t\right)}I+i\;\mathbf {\hat {n}} \cdot {\boldsymbol {\sigma }}\sin {\left(\omega t\right)}} . Physically, this corresponds to

6804-401: Is the requirement that the energies are real that implies the hermiticity of H {\displaystyle \mathbf {H} } . The eigenvectors represent the stationary states , i.e., those for whom the absolute magnitude of the squares of the probability amplitudes do not change with time. The most general form of a 2×2 Hermitian matrix such as the Hamiltonian of a two-state system

6930-432: Is the statevector at t = 0 {\displaystyle t=0} . Here the exponential of a matrix may be found from the series expansion. The matrix U ( t ) {\displaystyle \mathbf {U} (t)} is called the time evolution matrix (which comprises the matrix elements of the corresponding time evolution operator U ( t ) {\displaystyle U(t)} ). It

7056-431: Is usually ℏ {\textstyle \hbar } rather than h {\textstyle h} that gives the most reliable results when used in order-of-magnitude estimates . For example, using dimensional analysis to estimate the ionization energy of a hydrogen atom, the relevant parameters that determine the ionization energy E i {\textstyle E_{\text{i}}} are

7182-405: Is valid is when the system under consideration has two levels that are effectively decoupled from the system. This is the case in the analysis of the spontaneous or stimulated emission of light by atoms and that of charge qubits . In this case it should be kept in mind that the perturbations (interactions with an external field) are in the right range and do not cause transitions to states other than

7308-411: The Bloch vector precessing around n ^ {\displaystyle \mathbf {\hat {n}} } with angular frequency 2 ω {\displaystyle 2\omega } . Without loss of generality, assume the field is uniform and points in z ^ {\displaystyle \mathbf {\hat {z}} } , so that the time evolution operator

7434-506: The position operator x ^ {\displaystyle {\hat {x}}} and the momentum operator p ^ {\displaystyle {\hat {p}}} : where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta . The Planck relation connects the particular photon energy E with its associated wave frequency f : This energy

7560-399: The " ultraviolet catastrophe ", a name coined by Paul Ehrenfest in 1911. They contributed greatly (along with Einstein's work on the photoelectric effect ) in convincing physicists that Planck's postulate of quantized energy levels was more than a mere mathematical formalism. The first Solvay Conference in 1911 was devoted to "the theory of radiation and quanta". The photoelectric effect

7686-471: The "Dirac h {\textstyle h} " (or "Dirac's h {\textstyle h} " ). The combination h / ( 2 π ) {\textstyle h/(2\pi )} appeared in Niels Bohr 's 1913 paper, where it was denoted by M 0 {\textstyle M_{0}} . For the next 15 years, the combination continued to appear in

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7812-401: The "energy element" must be proportional to the frequency of the oscillator, the first version of what is now sometimes termed the " Planck–Einstein relation ": Planck was able to calculate the value of h {\displaystyle h} from experimental data on black-body radiation: his result, 6.55 × 10  J⋅s , is within 1.2% of the currently defined value. He also made

7938-587: The Bloch sphere will simply be R = ( cos ⁡ 2 ω t , − sin ⁡ 2 ω t , 0 ) {\displaystyle \mathbf {R} =\left(\cos {2\omega t},-\sin {2\omega t},0\right)} . This is a unit vector that begins pointing along x ^ {\displaystyle \mathbf {\hat {x}} } and precesses around z ^ {\displaystyle \mathbf {\hat {z}} } in

8064-459: The Hamiltonian is diagonal, i.e. | r | = δ {\displaystyle |\mathbf {r} |=\delta } and is of the form, H = ( E 1 0 0 E 2 ) . {\displaystyle \mathbf {H} ={\begin{pmatrix}E_{1}&0\\0&E_{2}\end{pmatrix}}.} Now, the unitary time evolution operator U {\displaystyle U}

8190-566: The Planck constant reflects the fact that everyday objects and systems are made of a large number of microscopic particles. For example, in green light (with a wavelength of 555  nanometres or a frequency of 540 THz ) each photon has an energy E = hf = 3.58 × 10  J . That is a very small amount of energy in terms of everyday experience, but everyday experience is not concerned with individual photons any more than with individual atoms or molecules. An amount of light more typical in everyday experience (though much larger than

8316-406: The actual proof that relativity was real. Before Einstein's paper, electromagnetic radiation such as visible light was considered to behave as a wave: hence the use of the terms "frequency" and "wavelength" to characterize different types of radiation. The energy transferred by a wave in a given time is called its intensity . The light from a theatre spotlight is more intense than the light from

8442-459: The advancement of Physics by his discovery of energy quanta". In metrology , the Planck constant is used, together with other constants, to define the kilogram , the SI unit of mass. The SI units are defined in such a way that, when the Planck constant is expressed in SI units, it has the exact value h {\displaystyle h} = 6.626 070 15 × 10  J⋅Hz . Planck's constant

8568-444: The amount of energy it emits at different radiation frequencies. It is the power emitted per unit area of the body, per unit solid angle of emission, per unit frequency. The spectral radiance can also be expressed per unit wavelength λ {\displaystyle \lambda } instead of per unit frequency. Substituting ν = c / λ {\displaystyle \nu =c/\lambda } in

8694-427: The analysis of the time evolution of the system and is easier to use with other specialized representations such as the Bloch sphere . If the two-state system's time-independent Hamiltonian H is defined as above, then its eigenvalues are given by E ± = α ± | r | {\displaystyle E_{\pm }=\alpha \pm |\mathbf {r} |} . Evidently, α

8820-622: The articles on Rabi cycle and rotating wave approximation . Consider the case of a spin-1/2 particle in a magnetic field B = B n ^ {\displaystyle \mathbf {B} =B\mathbf {\hat {n}} } . The interaction Hamiltonian for this system is H = − μ ⋅ B = − μ σ ⋅ B , {\displaystyle H=-{\boldsymbol {\mu }}\cdot \mathbf {B} =-\mu {\boldsymbol {\sigma }}\cdot \mathbf {B} ,} where μ {\displaystyle \mu }

8946-616: The basis states are orthonormal , ⟨ i | j ⟩ = δ i j {\displaystyle \langle i|j\rangle =\delta _{ij}} where i , j ∈ 1 , 2 {\displaystyle i,j\in {1,2}} and δ i j {\displaystyle \delta _{ij}} is the Kronecker delta , so c i = ⟨ i | ψ ⟩ {\displaystyle c_{i}=\langle i|\psi \rangle } . These two complex numbers may be considered coordinates in

9072-575: The basis states are not. The Time-dependent Schrödinger equation states i ℏ ∂ t | ψ ⟩ = H | ψ ⟩ {\textstyle i\hbar \partial _{t}|\psi \rangle =H|\psi \rangle } , and proceeding as before (substituting for | ψ ⟩ {\displaystyle |\psi \rangle } and premultiplying by ⟨ 1 | , ⟨ 2 | {\displaystyle \langle 1|,\langle 2|} again produces

9198-830: The basis states correspond to the basis vectors, | 1 ⟩ ≡ ( ⟨ 1 | 1 ⟩ ⟨ 2 | 1 ⟩ ) = ( 1 0 ) {\displaystyle |1\rangle \equiv {\begin{pmatrix}\langle 1|1\rangle \\\langle 2|1\rangle \end{pmatrix}}={\begin{pmatrix}1\\0\end{pmatrix}}} and | 2 ⟩ ≡ ( ⟨ 1 | 2 ⟩ ⟨ 2 | 2 ⟩ ) = ( 0 1 ) . {\displaystyle |2\rangle \equiv {\begin{pmatrix}\langle 1|2\rangle \\\langle 2|2\rangle \end{pmatrix}}={\begin{pmatrix}0\\1\end{pmatrix}}.} If

9324-902: The basis states from above, and multiplying both sides by ⟨ 1 | {\displaystyle \langle 1|} or ⟨ 2 | {\displaystyle \langle 2|} produces a system of two linear equations that can be written in matrix form, ( H 11 H 12 H 12 ∗ H 22 ) ( c 1 c 2 ) = E ( c 1 c 2 ) , {\displaystyle {\begin{pmatrix}H_{11}&H_{12}\\H_{12}^{*}&H_{22}\end{pmatrix}}{\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}}=E{\begin{pmatrix}c_{1}\\c_{2}\end{pmatrix}},} or H c = E c {\displaystyle \mathbf {Hc} =E\mathbf {c} } which

9450-887: The basis to the eigenvectors of the Hamiltonian, in other words, if the basis states | 1 ⟩ , | 2 ⟩ {\displaystyle |1\rangle ,|2\rangle } are chosen to be the eigenvectors, then ϵ 1 = H 11 = ⟨ 1 | H | 1 ⟩ = E 1 ⟨ 1 | 1 ⟩ = E 1 {\displaystyle \epsilon _{1}=H_{11}=\langle 1|H|1\rangle =E_{1}\langle 1|1\rangle =E_{1}} and β + i γ = H 21 = ⟨ 2 | H | 1 ⟩ = E 1 ⟨ 2 | 1 ⟩ = 0 {\displaystyle \beta +i\gamma =H_{21}=\langle 2|H|1\rangle =E_{1}\langle 2|1\rangle =0} and so

9576-780: The case of energy and the corresponding Hamiltonian , H , this means H i j = ⟨ i | H | j ⟩ = ⟨ j | H | i ⟩ ∗ = H j i ∗ , {\displaystyle H_{ij}=\langle i|H|j\rangle =\langle j|H|i\rangle ^{*}=H_{ji}^{*},} i.e. H 11 {\displaystyle H_{11}} and H 22 {\displaystyle H_{22}} are real, and H 12 = H 21 ∗ {\displaystyle H_{12}=H_{21}^{*}} . Thus, these four matrix elements H i j {\displaystyle H_{ij}} produce

9702-849: The collapsed section below), it can be shown that the Schrödinger equation becomes ∂ ψ ∂ t = i ( ω 1 σ x + ( w 0 + ω r 2 ) σ z ) ψ , {\displaystyle {\frac {\partial \psi }{\partial t}}=i\left(\omega _{1}\sigma _{x}+\left(w_{0}+{\frac {\omega _{r}}{2}}\right)\sigma _{z}\right)\psi ,} where ω 0 = μ B 0 / ℏ {\displaystyle \omega _{0}=\mu B_{0}/\hbar } and ω 1 = μ B 1 / ℏ {\displaystyle \omega _{1}=\mu B_{1}/\hbar } . As per

9828-526: The development of Niels Bohr 's atomic model and Bohr quoted him in his 1913 paper of the Bohr model of the atom. Bohr's model went beyond Planck's abstract harmonic oscillator concept: an electron in a Bohr atom could only have certain defined energies E n {\displaystyle E_{n}} where c {\displaystyle c} is the speed of light in vacuum, R ∞ {\displaystyle R_{\infty }}

9954-699: The dot product and dividing by i ℏ {\displaystyle i\hbar } yields ∂ ψ ∂ t = i ( ω 1 σ x cos ⁡ ω r t + ω 1 σ y sin ⁡ ω r t + ω 0 σ z ) ψ . {\displaystyle {\frac {\partial \psi }{\partial t}}=i\left(\omega _{1}\sigma _{x}\cos {\omega _{r}t}+\omega _{1}\sigma _{y}\sin {\omega _{r}t}+\omega _{0}\sigma _{z}\right)\psi .} To remove

10080-401: The dynamics of a spin in a magnetic field. An ideal magnet consists of a collection of identical spins behaving independently, and thus the total magnetization M {\displaystyle \mathbf {M} } is proportional to the Bloch vector R {\displaystyle \mathbf {R} } . All that is left to obtain the final form of the optical Bloch equations is

10206-469: The effect in terms of light quanta would earn him the Nobel Prize in 1921, after his predictions had been confirmed by the experimental work of Robert Andrews Millikan . The Nobel committee awarded the prize for his work on the photo-electric effect, rather than relativity, both because of a bias against purely theoretical physics not grounded in discovery or experiment, and dissent amongst its members as to

10332-488: The electrons in his model Bohr introduced the quantity h 2 π {\displaystyle {\frac {h}{2\pi }}} , now known as the reduced Planck constant as the quantum of angular momentum . The Planck constant also occurs in statements of Werner Heisenberg 's uncertainty principle. Given numerous particles prepared in the same state, the uncertainty in their position, Δ x {\displaystyle \Delta x} , and

10458-420: The energies of the individual states, which are by definition different). Upon setting c 1 {\displaystyle c_{1}} or c 2 {\displaystyle c_{2}} to be 0, only one state remains, and E {\displaystyle E} is the energy of the surviving state. This result is a redundant reminder that the time-independent Schrödinger equation

10584-399: The equations of motion for light describe a set of harmonic oscillators , one for each possible frequency. He examined how the entropy of the oscillators varied with the temperature of the body, trying to match Wien's law, and was able to derive an approximate mathematical function for the black-body spectrum, which gave a simple empirical formula for long wavelengths. Planck tried to find

10710-446: The exact solution to a time dependent Hamiltonian. The NMR phenomenon is achieved by placing a nucleus in a strong, static field B 0 (the "holding field") and then applying a weak, transverse field B 1 that oscillates at some radiofrequency ω r . Explicitly, consider a spin-1/2 particle in a holding field B 0 z ^ {\displaystyle B_{0}\mathbf {\hat {z}} } and

10836-496: The expectation value of each Pauli matrix is a component of the Bloch vector , ⟨ σ i ⟩ = ψ † σ i ψ = R i {\displaystyle \langle \sigma _{i}\rangle =\psi ^{\dagger }\sigma _{i}\psi =R_{i}} . Equating the left and right hand sides, and noting that 2 μ ℏ {\displaystyle {\frac {2\mu }{\hbar }}}

10962-460: The fact that R i = ⟨ σ i ⟩ {\displaystyle \mathbf {R} _{i}=\langle \sigma _{i}\rangle } , this equation is the same equation as before. Two-state systems are the simplest non-trivial quantum systems that occur in nature, but the above-mentioned methods of analysis are not just valid for simple two-state systems. Any general multi-state quantum system can be treated as

11088-529: The fact that ε i j k = ε k i j {\displaystyle \varepsilon _{ijk}=\varepsilon _{kij}} has been used. In vector form these three equations can be expressed in terms of a cross product ∂ R ∂ t = γ R × B {\displaystyle {\frac {\partial \mathbf {R} }{\partial t}}=\gamma \mathbf {R} \times \mathbf {B} } Classically, this equation describes

11214-410: The first determination of the Boltzmann constant k B {\displaystyle k_{\text{B}}} from the same data and theory. The black-body problem was revisited in 1905, when Lord Rayleigh and James Jeans (together) and Albert Einstein independently proved that classical electromagnetism could never account for the observed spectrum. These proofs are commonly known as

11340-418: The form of detectable photons. This is the fundamental basis for NMR , and in practice is accomplished by scanning ω r {\displaystyle \omega _{r}} until the resonant frequency is found at which point the sample will emit light. Similar calculations are done in atomic physics, and in the case that the field is not rotating, but oscillating with a complex amplitude, use

11466-592: The free precession case, the Hamiltonian is H = − μ σ ⋅ B {\displaystyle H=-\mu {\boldsymbol {\sigma }}\cdot \mathbf {B} } , and the evolution of a state vector ψ ( t ) {\displaystyle \psi (t)} is found by solving the time-dependent Schrödinger equation H ψ = i ℏ ∂ ψ / ∂ t {\displaystyle H\psi =i\hbar \,\partial \psi /\partial t} . After some manipulation (given in

11592-462: The frequency of incident light f {\displaystyle f} and the kinetic energy of photoelectrons E {\displaystyle E} was shown to be equal to the Planck constant h {\displaystyle h} . In 1912 John William Nicholson developed an atomic model and found the angular momentum of the electrons in the model were related by h /2 π . Nicholson's nuclear quantum atomic model influenced

11718-1248: The inclusion of the phenomenological relaxation terms. As a final aside, the above equation can be derived by considering the time evolution of the angular momentum operator in the Heisenberg picture . i ℏ d σ j d t = [ σ j , H ] = [ σ j , − μ σ i B i ] = − μ ( σ j σ i B i − σ i σ j B i ) = μ [ σ i , σ j ] B i = 2 μ i ε i j k σ k B i {\displaystyle i\hbar {\frac {d\sigma _{j}}{dt}}=\left[\sigma _{j},H\right]=\left[\sigma _{j},-\mu \sigma _{i}B_{i}\right]=-\mu \left(\sigma _{j}\sigma _{i}B_{i}-\sigma _{i}\sigma _{j}B_{i}\right)=\mu [\sigma _{i},\sigma _{j}]B_{i}=2\mu i\varepsilon _{ijk}\sigma _{k}B_{i}} When coupled with

11844-440: The individual eigenstate energies still inside the product vector. In either case, the Hamiltonian matrix can be derived using the method specified above, or via the more traditional method of constructing a matrix using boundary conditions; specifically, by using the requirement that when it acts on either basis state, it must return that state multiplied by the energy of that state. (There are no boundary conditions on how it acts on

11970-510: The initial condition. The frequency Ω = | r | ℏ = 1 ℏ β 2 + γ 2 + δ 2 = | Ω R | 2 + Δ 2 {\displaystyle \Omega ={\frac {|\mathbf {r} |}{\hbar }}={\frac {1}{\hbar }}{\sqrt {\beta ^{2}+\gamma ^{2}+\delta ^{2}}}={\sqrt {|\Omega _{R}|^{2}+\Delta ^{2}}}}

12096-423: The interaction is given by an appropriate coupling term that is analogous to the magnetic moment. The precession of the state vector (which need not be a physical spinning as in the previous case) can be viewed as the precession of the state vector on the Bloch sphere . The representation on the Bloch sphere for a state vector ψ ( 0 ) {\displaystyle \psi (0)} will simply be

12222-419: The literature, but normally without a separate symbol. Then, in 1926, in their seminal papers, Schrödinger and Dirac again introduced special symbols for it: K {\textstyle K} in the case of Schrödinger, and h {\textstyle h} in the case of Dirac. Dirac continued to use h {\textstyle h} in this way until 1930, when he introduced

12348-687: The mass of the electron m e {\textstyle m_{\text{e}}} , the electron charge e {\textstyle e} , and either the Planck constant h {\textstyle h} or the reduced Planck constant ℏ {\textstyle \hbar } : E i ∝ m e e 4 / h 2   or   ∝ m e e 4 / ℏ 2 {\displaystyle E_{\text{i}}\propto m_{\text{e}}e^{4}/h^{2}\ {\text{or}}\ \propto m_{\text{e}}e^{4}/\hbar ^{2}} Since both constants have

12474-505: The new eigenvectors of the perturbed system can be solved for exactly, as mentioned in the introduction. Suppose that the system starts in one of the basis states at t = 0 {\displaystyle t=0} , say | 1 ⟩ {\displaystyle |1\rangle } so that c 0 = ( 1 0 ) {\textstyle \mathbf {c} _{0}={\begin{pmatrix}1\\0\end{pmatrix}}} , and we are interested in

12600-465: The ones of interest. Pedagogically, the two-state formalism is among the simplest of mathematical techniques used for the analysis of quantum systems. It can be used to illustrate fundamental quantum mechanical phenomena such as the interference exhibited by particles of the polarization states of the photon, but also more complex phenomena such as neutrino oscillation or the neutral K-meson oscillation. Reduced Planck constant The constant

12726-461: The overall phase of the operator, and can usually be ignored to yield a new time evolution operator that is physically indistinguishable from the original operator. Moreover, any perturbation to the system (which will be of the same form as the Hamiltonian) can be added to the system in the eigenbasis of the unperturbed Hamiltonian and analysed in the same way as above. Therefore, for any perturbation

12852-408: The particle is represented by a wavefunction spread out in space and in time. Related to this is the concept of energy quantization which existed in old quantum theory and also exists in altered form in modern quantum physics. Classical physics cannot explain quantization of energy. The Planck constant has the same dimensions as action and as angular momentum . In SI units, the Planck constant

12978-427: The previous section, the solution to this equation has the Bloch vector precessing around ( ω 1 , 0 , ω 0 + ω r / 2 ) {\displaystyle (\omega _{1},0,\omega _{0}+\omega _{r}/2)} with a frequency that is twice the magnitude of the vector. If ω 0 {\displaystyle \omega _{0}}

13104-903: The probability of occupation of each of the basis states as a function of time when H {\displaystyle \mathbf {H} } is the time-independent Hamiltonian. c ( t ) = U ( t ) c 0 = ( U 11 ( t ) U 12 ( t ) U 21 ( t ) U 22 ( t ) ) ( 1 0 ) = ( U 11 ( t ) U 21 ( t ) ) . {\displaystyle \mathbf {c} (t)=\mathbf {U} (t)\mathbf {c} _{0}={\begin{pmatrix}U_{11}(t)&U_{12}(t)\\U_{21}(t)&U_{22}(t)\end{pmatrix}}{\begin{pmatrix}1\\0\end{pmatrix}}={\begin{pmatrix}U_{11}(t)\\U_{21}(t)\end{pmatrix}}.} The probability of occupation of state i

13230-448: The proportionality between the momentum and the quantum wavelength of not just the photon, but the quantum wavelength of any particle. This was confirmed by experiments soon afterward. This holds throughout the quantum theory, including electrodynamics . The de Broglie wavelength λ of the particle is given by where p denotes the linear momentum of a particle, such as a photon, or any other elementary particle . The energy of

13356-417: The relation above we get showing how radiated energy emitted at shorter wavelengths increases more rapidly with temperature than energy emitted at longer wavelengths. Planck's law may also be expressed in other terms, such as the number of photons emitted at a certain wavelength, or the energy density in a volume of radiation. The SI unit of B ν {\displaystyle B_{\nu }}

13482-5216: The right hand side of the equation e i σ z ω r t / 2 σ x e − i σ z ω r t / 2 = ( e i ω r t / 2 0 0 e − i ω r t / 2 ) ( 0 1 1 0 ) ( e − i ω r t / 2 0 0 e i ω r t / 2 ) = ( 0 e i ω r t e − i ω r t 0 ) {\displaystyle e^{i\sigma _{z}\omega _{r}t/2}\sigma _{x}e^{-i\sigma _{z}\omega _{r}t/2}={\begin{pmatrix}e^{i\omega _{r}t/2}&0\\0&e^{-i\omega _{r}t/2}\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}e^{-i\omega _{r}t/2}&0\\0&e^{i\omega _{r}t/2}\end{pmatrix}}={\begin{pmatrix}0&e^{i\omega _{r}t}\\e^{-i\omega _{r}t}&0\end{pmatrix}}} e i σ z ω r t / 2 σ y e − i σ z ω r t / 2 = ( e i ω r t / 2 0 0 e − i ω r t / 2 ) ( 0 − i i 0 ) ( e − i ω r t / 2 0 0 e i ω r t / 2 ) = ( 0 − i e i ω r t i e − i ω r t 0 ) {\displaystyle e^{i\sigma _{z}\omega _{r}t/2}\sigma _{y}e^{-i\sigma _{z}\omega _{r}t/2}={\begin{pmatrix}e^{i\omega _{r}t/2}&0\\0&e^{-i\omega _{r}t/2}\end{pmatrix}}{\begin{pmatrix}0&-i\\i&0\end{pmatrix}}{\begin{pmatrix}e^{-i\omega _{r}t/2}&0\\0&e^{i\omega _{r}t/2}\end{pmatrix}}={\begin{pmatrix}0&-ie^{i\omega _{r}t}\\ie^{-i\omega _{r}t}&0\end{pmatrix}}} e i σ z ω r t / 2 σ z e − i σ z ω r t / 2 = ( e i ω r t / 2 0 0 e − i ω r t / 2 ) ( 1 0 0 − 1 ) ( e − i ω r t / 2 0 0 e i ω r t / 2 ) = σ z {\displaystyle e^{i\sigma _{z}\omega _{r}t/2}\sigma _{z}e^{-i\sigma _{z}\omega _{r}t/2}={\begin{pmatrix}e^{i\omega _{r}t/2}&0\\0&e^{-i\omega _{r}t/2}\end{pmatrix}}{\begin{pmatrix}1&0\\0&-1\end{pmatrix}}{\begin{pmatrix}e^{-i\omega _{r}t/2}&0\\0&e^{i\omega _{r}t/2}\end{pmatrix}}=\sigma _{z}} The equation now reads ∂ ψ ∂ t = i ( ω 1 ( 0 e i ω r t ( cos ⁡ ω r t − i sin ⁡ ω r t ) e − i ω r t ( cos ⁡ ω r t + i sin ⁡ ω r t ) 0 ) + ( w 0 + ω r 2 ) σ z ) ψ , {\displaystyle {\frac {\partial \psi }{\partial t}}=i\left(\omega _{1}{\begin{pmatrix}0&e^{i\omega _{r}t}\left(\cos {\omega _{r}t}-i\sin {\omega _{r}t}\right)\\e^{-i\omega _{r}t}\left(\cos {\omega _{r}t}+i\sin {\omega _{r}t}\right)&0\end{pmatrix}}+\left(w_{0}+{\frac {\omega _{r}}{2}}\right)\sigma _{z}\right)\psi ,} which by Euler's identity becomes ∂ ψ ∂ t = i ( ω 1 σ x + ( w 0 + ω r 2 ) σ z ) ψ {\displaystyle {\frac {\partial \psi }{\partial t}}=i\left(\omega _{1}\sigma _{x}+\left(w_{0}+{\frac {\omega _{r}}{2}}\right)\sigma _{z}\right)\psi } The optical Bloch equations for

13608-413: The same dimensions, they will enter the dimensional analysis in the same way, but with ℏ {\textstyle \hbar } the estimate is within a factor of two, while with h {\textstyle h} the error is closer to ( 2 π ) 2 ≈ 40 {\textstyle (2\pi )^{2}\approx 40} . The reduced Planck constant

13734-415: The same term [REDACTED] This disambiguation page lists articles associated with the title Two-state . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Two-state&oldid=1251164176 " Category : Disambiguation pages Hidden categories: Short description

13860-840: The smallest amount perceivable by the human eye) is the energy of one mole of photons; its energy can be computed by multiplying the photon energy by the Avogadro constant , N A  =  6.022 140 76 × 10  mol ‍ , with the result of 216 kJ , about the food energy in three apples. Many equations in quantum physics are customarily written using the reduced Planck constant , equal to h / ( 2 π ) {\textstyle h/(2\pi )} and denoted ℏ {\textstyle \hbar } (pronounced h-bar ). The fundamental equations look simpler when written using ℏ {\textstyle \hbar } as opposed to h {\textstyle h} , and it

13986-454: The state | ψ ⟩ {\displaystyle |\psi \rangle } is normalized , the norm of the state vector is unity, i.e. | c 1 | 2 + | c 2 | 2 = 1 {\displaystyle {|c_{1}|}^{2}+{|c_{2}|}^{2}=1} . All observable physical quantities , such as energy, are associated with hermitian operators . In

14112-422: The state can be written as a superposition of these two states with probability amplitudes c 1 , c 2 {\displaystyle c_{1},c_{2}} , | ψ ⟩ = c 1 | 1 ⟩ + c 2 | 2 ⟩ . {\displaystyle |\psi \rangle =c_{1}|1\rangle +c_{2}|2\rangle .} Since

14238-426: The system to interact with the field of a particular direction and strength for precise durations, it is possible to obtain any orientation of the Bloch vector , which is equivalent to obtaining any complex superposition. This is the basis for numerous technologies including quantum computing and MRI . Nuclear magnetic resonance (NMR) is an important example in the dynamics of two-state systems because it involves

14364-1389: The system, namely the eigenvalues of the Hamiltonian matrix, can be found in the usual way. Equivalently, this matrix can be decomposed as, H = α ⋅ σ 0 + β ⋅ σ 1 + γ ⋅ σ 2 + δ ⋅ σ 3 = ( α + δ β − i γ β + i γ α − δ ) . {\displaystyle \mathbf {H} =\alpha \cdot \sigma _{0}+\beta \cdot \sigma _{1}+\gamma \cdot \sigma _{2}+\delta \cdot \sigma _{3}={\begin{pmatrix}\alpha +\delta &\beta -i\gamma \\\beta +i\gamma &\alpha -\delta \end{pmatrix}}.} Here, α = 1 2 ( ε 1 + ε 2 ) {\textstyle \alpha ={\frac {1}{2}}\left(\varepsilon _{1}+\varepsilon _{2}\right)} and δ = 1 2 ( ε 1 − ε 2 ) {\textstyle \delta ={\frac {1}{2}}\left(\varepsilon _{1}-\varepsilon _{2}\right)} are real numbers. The matrix σ 0 {\displaystyle \sigma _{0}}

14490-2487: The time dependence from the problem, the wave function is transformed according to ψ → e − i σ z ω r t / 2 ψ {\displaystyle \psi \rightarrow e^{-i\sigma _{z}\omega _{r}t/2}\psi } . The time dependent Schrödinger equation becomes − i σ z ω r 2 e − i σ z ω r t / 2 ψ + e − i σ z ω r t / 2 ∂ ψ ∂ t = i ( ω 1 σ x cos ⁡ ω r t + ω 1 σ y sin ⁡ ω r t + ω 0 σ z ) e − i σ z ω r t / 2 ψ , {\displaystyle -i\sigma _{z}{\frac {\omega _{r}}{2}}e^{-i\sigma _{z}\omega _{r}t/2}\psi +e^{-i\sigma _{z}\omega _{r}t/2}{\frac {\partial \psi }{\partial t}}=i\left(\omega _{1}\sigma _{x}\cos {\omega _{r}t}+\omega _{1}\sigma _{y}\sin {\omega _{r}t}+\omega _{0}\sigma _{z}\right)e^{-i\sigma _{z}\omega _{r}t/2}\psi ,} which after some rearrangement yields ∂ ψ ∂ t = i e i σ z ω r t / 2 ( ω 1 σ x cos ⁡ ω r t + ω 1 σ y sin ⁡ ω r t + ( ω 0 + ω r 2 ) σ z ) e − i σ z ω r t / 2 ψ {\displaystyle {\frac {\partial \psi }{\partial t}}=ie^{i\sigma _{z}\omega _{r}t/2}\left(\omega _{1}\sigma _{x}\cos {\omega _{r}t}+\omega _{1}\sigma _{y}\sin {\omega _{r}t}+\left(\omega _{0}+{\frac {\omega _{r}}{2}}\right)\sigma _{z}\right)e^{-i\sigma _{z}\omega _{r}t/2}\psi } Evaluating each term on

14616-407: The uncertainty in their momentum, Δ p x {\displaystyle \Delta p_{x}} , obey where the uncertainty is given as the standard deviation of the measured value from its expected value . There are several other such pairs of physically measurable conjugate variables which obey a similar rule. One example is time vs. energy. The inverse relationship between

14742-403: The uncertainty of the two conjugate variables forces a tradeoff in quantum experiments, as measuring one quantity more precisely results in the other quantity becoming imprecise. In addition to some assumptions underlying the interpretation of certain values in the quantum mechanical formulation, one of the fundamental cornerstones to the entire theory lies in the commutator relationship between

14868-451: The unit J⋅s, which is equal to kg⋅m ⋅s , where the metre and the second are defined in terms of speed of light c and duration of hyperfine transition of the ground state of an unperturbed caesium-133 atom Δ ν Cs ." Technologies of mass metrology such as the Kibble balance measure refine the value of kilogram applying fixed value of the Planck constant. The Planck constant

14994-404: The vector of expectation values R = ( ⟨ σ x ⟩ , ⟨ σ y ⟩ , ⟨ σ z ⟩ ) {\displaystyle \mathbf {R} =\left(\langle \sigma _{x}\rangle ,\langle \sigma _{y}\rangle ,\langle \sigma _{z}\rangle \right)} . As an example, consider

15120-458: The wave description of light. The "photoelectrons" emitted as a result of the photoelectric effect have a certain kinetic energy , which can be measured. This kinetic energy (for each photoelectron) is independent of the intensity of the light, but depends linearly on the frequency; and if the frequency is too low (corresponding to a photon energy that is less than the work function of the material), no photoelectrons are emitted at all, unless

15246-556: Was to interpret U N [ the vibrational energy of N oscillators ] not as a continuous, infinitely divisible quantity, but as a discrete quantity composed of an integral number of finite equal parts. Let us call each such part the energy element ε ; With this new condition, Planck had imposed the quantization of the energy of the oscillators, "a purely formal assumption ... actually I did not think much about it ..." in his own words, but one that would revolutionize physics. Applying this new approach to Wien's displacement law showed that

15372-533: Was formulated as part of Max Planck's successful effort to produce a mathematical expression that accurately predicted the observed spectral distribution of thermal radiation from a closed furnace ( black-body radiation ). This mathematical expression is now known as Planck's law. In the last years of the 19th century, Max Planck was investigating the problem of black-body radiation first posed by Kirchhoff some 40 years earlier. Every physical body spontaneously and continuously emits electromagnetic radiation . There

15498-548: Was no expression or explanation for the overall shape of the observed emission spectrum. At the time, Wien's law fit the data for short wavelengths and high temperatures, but failed for long wavelengths. Also around this time, but unknown to Planck, Lord Rayleigh had derived theoretically a formula, now known as the Rayleigh–Jeans law , that could reasonably predict long wavelengths but failed dramatically at short wavelengths. Approaching this problem, Planck hypothesized that

15624-467: Was postulated by Max Planck in 1900 as a proportionality constant needed to explain experimental black-body radiation. Planck later referred to the constant as the "quantum of action ". In 1905, Albert Einstein associated the "quantum" or minimal element of the energy to the electromagnetic wave itself. Max Planck received the 1918 Nobel Prize in Physics "in recognition of the services he rendered to

15750-661: Was speculated that physical action could not take on an arbitrary value, but instead was restricted to integer multiples of a very small quantity, the "[elementary] quantum of action", now called the Planck constant . This was a significant conceptual part of the so-called " old quantum theory " developed by physicists including Bohr , Sommerfeld , and Ishiwara , in which particle trajectories exist but are hidden , but quantum laws constrain them based on their action. This view has been replaced by fully modern quantum theory, in which definite trajectories of motion do not even exist; rather,

15876-436: Was that light itself is quantized; that the energy of light is not transferred continuously as in a classical wave, but only in small "packets" or quanta. The size of these "packets" of energy, which would later be named photons , was to be the same as Planck's "energy element", giving the modern version of the Planck–Einstein relation: Einstein's postulate was later proven experimentally: the constant of proportionality between

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