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123-422: The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator . Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point , it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution

246-475: A {\displaystyle a} larger we make the spread in momentum smaller, but the spread in position gets larger. This illustrates the uncertainty principle. As we let the Gaussian wave packet evolve in time, we see that its center moves through space at a constant velocity (like a classical particle with no forces acting on it). However, the wave packet will also spread out as time progresses, which means that

369-524: A † = m ω 2 ℏ ( x ^ − i m ω p ^ ) {\displaystyle {\begin{aligned}a&={\sqrt {m\omega \over 2\hbar }}\left({\hat {x}}+{i \over m\omega }{\hat {p}}\right)\\a^{\dagger }&={\sqrt {m\omega \over 2\hbar }}\left({\hat {x}}-{i \over m\omega }{\hat {p}}\right)\end{aligned}}} Note these operators classically are exactly

492-486: A † e − α ∗ a | 0 ⟩ . {\displaystyle |\alpha \rangle =\sum _{n=0}^{\infty }|n\rangle \langle n|\alpha \rangle =e^{-{\frac {1}{2}}|\alpha |^{2}}\sum _{n=0}^{\infty }{\frac {\alpha ^{n}}{\sqrt {n!}}}|n\rangle =e^{-{\frac {1}{2}}|\alpha |^{2}}e^{\alpha a^{\dagger }}e^{-{\alpha ^{*}a}}|0\rangle .} Since coherent states are not energy eigenstates, their time evolution

615-939: A † | n − 1 ⟩ = 1 n ( n − 1 ) ( a † ) 2 | n − 2 ⟩ = ⋯ = 1 n ! ( a † ) n | 0 ⟩ . {\displaystyle {\begin{aligned}\langle n|aa^{\dagger }|n\rangle &=\langle n|\left([a,a^{\dagger }]+a^{\dagger }a\right)\left|n\right\rangle =\langle n|\left(N+1\right)|n\rangle =n+1\\[1ex]\Rightarrow a^{\dagger }|n\rangle &={\sqrt {n+1}}|n+1\rangle \\[1ex]\Rightarrow |n\rangle &={\frac {1}{\sqrt {n}}}a^{\dagger }\left|n-1\right\rangle ={\frac {1}{\sqrt {n(n-1)}}}\left(a^{\dagger }\right)^{2}\left|n-2\right\rangle =\cdots ={\frac {1}{\sqrt {n!}}}\left(a^{\dagger }\right)^{n}\left|0\right\rangle .\end{aligned}}} The preceding analysis

738-689: A † | n ⟩ = ( a † N + [ N , a † ] ) | n ⟩ = ( a † N + a † ) | n ⟩ = ( n + 1 ) a † | n ⟩ , {\displaystyle {\begin{aligned}Na^{\dagger }|n\rangle &=\left(a^{\dagger }N+[N,a^{\dagger }]\right)|n\rangle \\&=\left(a^{\dagger }N+a^{\dagger }\right)|n\rangle \\&=(n+1)a^{\dagger }|n\rangle ,\end{aligned}}} and similarly, N

861-549: A † | n ⟩ = ⟨ n | ( [ a , a † ] + a † a ) | n ⟩ = ⟨ n | ( N + 1 ) | n ⟩ = n + 1 ⇒ a † | n ⟩ = n + 1 | n + 1 ⟩ ⇒ | n ⟩ = 1 n

984-655: A † ∣ 0 ⟩ = ψ 1 ( x )   , {\displaystyle \langle x\mid a^{\dagger }\mid 0\rangle =\psi _{1}(x)~,} so that ψ 1 ( x , t ) = ⟨ x ∣ e − 3 i ω t / 2 a † ∣ 0 ⟩ {\displaystyle \psi _{1}(x,t)=\langle x\mid e^{-3i\omega t/2}a^{\dagger }\mid 0\rangle } , and so on. The quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify

1107-436: A † + a ) p ^ = i ℏ m ω 2 ( a † − a )   . {\displaystyle {\begin{aligned}{\hat {x}}&={\sqrt {\frac {\hbar }{2m\omega }}}(a^{\dagger }+a)\\{\hat {p}}&=i{\sqrt {\frac {\hbar m\omega }{2}}}(a^{\dagger }-a)~.\end{aligned}}} The operator

1230-403: A † a | n ⟩ = ( a | n ⟩ ) † a | n ⟩ ⩾ 0 , {\displaystyle n=\langle n|N|n\rangle =\langle n|a^{\dagger }a|n\rangle ={\Bigl (}a|n\rangle {\Bigr )}^{\dagger }a|n\rangle \geqslant 0,} the smallest eigenvalue of the number operator is 0, and

1353-398: A | n ⟩ = ( n − 1 ) a | n ⟩ . {\displaystyle Na|n\rangle =(n-1)a|n\rangle .} This means that a acts on | n ⟩ to produce, up to a multiplicative constant, | n –1⟩ , and a acts on | n ⟩ to produce | n +1⟩ . For this reason, a is called an annihilation operator ("lowering operator"), and

SECTION 10

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1476-445: A | 0 ⟩ = 0 {\displaystyle a\left|0\right\rangle =0} and via the Kermack-McCrae identity, the last form is equivalent to a unitary displacement operator acting on the ground state: | α ⟩ = e α a ^ † − α ∗

1599-546: A | 0 ⟩ = 0. {\displaystyle a\left|0\right\rangle =0.} In this case, subsequent applications of the lowering operator will just produce zero, instead of additional energy eigenstates. Furthermore, we have shown above that H ^ | 0 ⟩ = ℏ ω 2 | 0 ⟩ {\displaystyle {\hat {H}}\left|0\right\rangle ={\frac {\hbar \omega }{2}}\left|0\right\rangle } Finally, by acting on |0⟩ with

1722-1179: A ^ | 0 ⟩ = D ( α ) | 0 ⟩ {\displaystyle |\alpha \rangle =e^{\alpha {\hat {a}}^{\dagger }-\alpha ^{*}{\hat {a}}}|0\rangle =D(\alpha )|0\rangle } . Calculating the expectation values: ⟨ x ^ ⟩ α ( t ) = 2 ℏ m ω | α 0 | cos ⁡ ( ω t − ϕ ) {\displaystyle \langle {\hat {x}}\rangle _{\alpha (t)}={\sqrt {\frac {2\hbar }{m\omega }}}|\alpha _{0}|\cos {(\omega t-\phi )}} ⟨ p ^ ⟩ α ( t ) = − 2 m ℏ ω | α 0 | sin ⁡ ( ω t − ϕ ) {\displaystyle \langle {\hat {p}}\rangle _{\alpha (t)}=-{\sqrt {2m\hbar \omega }}|\alpha _{0}|\sin {(\omega t-\phi )}} Quantum mechanics Quantum mechanics

1845-481: A a creation operator ("raising operator"). The two operators together are called ladder operators . Given any energy eigenstate, we can act on it with the lowering operator, a , to produce another eigenstate with ħω less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to E = −∞ . However, since n = ⟨ n | N | n ⟩ = ⟨ n |

1968-596: A is not Hermitian , since itself and its adjoint a are not equal. The energy eigenstates | n ⟩ , when operated on by these ladder operators, give a † | n ⟩ = n + 1 | n + 1 ⟩ a | n ⟩ = n | n − 1 ⟩ . {\displaystyle {\begin{aligned}a^{\dagger }|n\rangle &={\sqrt {n+1}}|n+1\rangle \\a|n\rangle &={\sqrt {n}}|n-1\rangle .\end{aligned}}} From

2091-839: A spectral method . It turns out that there is a family of solutions. In this basis, they amount to Hermite functions , ψ n ( x ) = 1 2 n n ! ( m ω π ℏ ) 1 / 4 e − m ω x 2 2 ℏ H n ( m ω ℏ x ) , n = 0 , 1 , 2 , … . {\displaystyle \psi _{n}(x)={\frac {1}{\sqrt {2^{n}\,n!}}}\left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}e^{-{\frac {m\omega x^{2}}{2\hbar }}}H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad n=0,1,2,\ldots .} The functions H n are

2214-475: A classical oscillator), but have a small range of variance, in accordance with the Heisenberg uncertainty principle . The ground state probability density is concentrated at the origin, which means the particle spends most of its time at the bottom of the potential well, as one would expect for a state with little energy. As the energy increases, the probability density peaks at the classical "turning points", where

2337-929: A classical system. They are eigenvectors of the annihilation operator, not the Hamiltonian, and form an overcomplete basis which consequentially lacks orthogonality. The coherent states are indexed by α ∈ C {\displaystyle \alpha \in \mathbb {C} } and expressed in the | n ⟩ basis as | α ⟩ = ∑ n = 0 ∞ | n ⟩ ⟨ n | α ⟩ = e − 1 2 | α | 2 ∑ n = 0 ∞ α n n ! | n ⟩ = e − 1 2 | α | 2 e α

2460-406: A definite prediction of what the quantum state ψ ( t ) {\displaystyle \psi (t)} will be at any later time. Some wave functions produce probability distributions that are independent of time, such as eigenstates of the Hamiltonian . Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example,

2583-510: A family of unitary operators parameterized by a variable t {\displaystyle t} . Under the evolution generated by A {\displaystyle A} , any observable B {\displaystyle B} that commutes with A {\displaystyle A} will be conserved. Moreover, if B {\displaystyle B} is conserved by evolution under A {\displaystyle A} , then A {\displaystyle A}

SECTION 20

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2706-541: A given initial configuration ψ ( x ,0) then is simply ψ ( x , t ) = ∫ d y   K ( x , y ; t ) ψ ( y , 0 ) . {\displaystyle \psi (x,t)=\int dy~K(x,y;t)\psi (y,0)\,.} The coherent states (also known as Glauber states) of the harmonic oscillator are special nondispersive wave packets , with minimum uncertainty σ x σ p = ℏ ⁄ 2 , whose observables ' expectation values evolve like

2829-471: A loss of information, though: knowing the reduced density matrices of the individual systems is not enough to reconstruct the state of the composite system. Just as density matrices specify the state of a subsystem of a larger system, analogously, positive operator-valued measures (POVMs) describe the effect on a subsystem of a measurement performed on a larger system. POVMs are extensively used in quantum information theory. As described above, entanglement

2952-426: A mathematical formulation of quantum mechanics and survey its application to some useful and oft-studied examples. In the mathematically rigorous formulation of quantum mechanics, the state of a quantum mechanical system is a vector ψ {\displaystyle \psi } belonging to a ( separable ) complex Hilbert space H {\displaystyle {\mathcal {H}}} . This vector

3075-417: A measurement of its position and also at the same time for a measurement of its momentum . Another consequence of the mathematical rules of quantum mechanics is the phenomenon of quantum interference , which is often illustrated with the double-slit experiment . In the basic version of this experiment, a coherent light source , such as a laser beam, illuminates a plate pierced by two parallel slits, and

3198-467: A probability amplitude. Applying the Born rule to these amplitudes gives a probability density function for the position that the electron will be found to have when an experiment is performed to measure it. This is the best the theory can do; it cannot say for certain where the electron will be found. The Schrödinger equation relates the collection of probability amplitudes that pertain to one moment of time to

3321-553: A real number (which needs to be determined) that will specify a time-independent energy level , or eigenvalue , and the solution | ψ ⟩ {\displaystyle |\psi \rangle } denotes that level's energy eigenstate . Then solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave function ⟨ x | ψ ⟩ = ψ ( x ) {\displaystyle \langle x|\psi \rangle =\psi (x)} , using

3444-405: A single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus , whereas in quantum mechanics, it is described by a static wave function surrounding the nucleus. For example, the electron wave function for an unexcited hydrogen atom is a spherically symmetric function known as an s orbital ( Fig. 1 ). Analytic solutions of

3567-545: A single spatial dimension. A free particle is one which is not subject to external influences, so that its Hamiltonian consists only of its kinetic energy: The general solution of the Schrödinger equation is given by which is a superposition of all possible plane waves e i ( k x − ℏ k 2 2 m t ) {\displaystyle e^{i(kx-{\frac {\hbar k^{2}}{2m}}t)}} , which are eigenstates of

3690-1040: Is ∇ ψ = e x ∂ ψ ∂ x + e y ∂ ψ ∂ y + e z ∂ ψ ∂ z = i ℏ ( p x e x + p y e y + p z e z ) ψ = i ℏ p ψ {\displaystyle {\begin{aligned}\nabla \psi &=\mathbf {e} _{x}{\frac {\partial \psi }{\partial x}}+\mathbf {e} _{y}{\frac {\partial \psi }{\partial y}}+\mathbf {e} _{z}{\frac {\partial \psi }{\partial z}}\\&={\frac {i}{\hbar }}\left(p_{x}\mathbf {e} _{x}+p_{y}\mathbf {e} _{y}+p_{z}\mathbf {e} _{z}\right)\psi \\&={\frac {i}{\hbar }}\mathbf {p} \psi \end{aligned}}} where e x , e y , and e z are

3813-468: Is and this provides the lower bound on the product of standard deviations: Another consequence of the canonical commutation relation is that the position and momentum operators are Fourier transforms of each other, so that a description of an object according to its momentum is the Fourier transform of its description according to its position. The fact that dependence in momentum is the Fourier transform of

Quantum harmonic oscillator - Misplaced Pages Continue

3936-428: Is a linear operator , the momentum operator is also linear, and because any wave function can be expressed as a superposition of other states, when this momentum operator acts on the entire superimposed wave, it yields the momentum eigenvalues for each plane wave component. These new components then superimpose to form the new state, in general not a multiple of the old wave function. The derivation in three dimensions

4059-469: Is a fundamental theory that describes the behavior of nature at and below the scale of atoms . It is the foundation of all quantum physics , which includes quantum chemistry , quantum field theory , quantum technology , and quantum information science . Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary ( macroscopic and (optical) microscopic ) scale, but

4182-478: Is a key feature of models of measurement processes in which an apparatus becomes entangled with the system being measured. Systems interacting with the environment in which they reside generally become entangled with that environment, a phenomenon known as quantum decoherence . This can explain why, in practice, quantum effects are difficult to observe in systems larger than microscopic. There are many mathematically equivalent formulations of quantum mechanics. One of

4305-424: Is a valid joint state that is not separable. States that are not separable are called entangled . If the state for a composite system is entangled, it is impossible to describe either component system A or system B by a state vector. One can instead define reduced density matrices that describe the statistics that can be obtained by making measurements on either component system alone. This necessarily causes

4428-688: Is algebraic, using only the commutation relations between the raising and lowering operators. Once the algebraic analysis is complete, one should turn to analytical questions. First, one should find the ground state, that is, the solution of the equation a ψ 0 = 0 {\displaystyle a\psi _{0}=0} . In the position representation, this is the first-order differential equation ( x + ℏ m ω d d x ) ψ 0 = 0 , {\displaystyle \left(x+{\frac {\hbar }{m\omega }}{\frac {d}{dx}}\right)\psi _{0}=0,} whose solution

4551-510: Is called minimal coupling . For electrically neutral particles, the canonical momentum is equal to the kinetic momentum. The momentum operator can be described as a symmetric (i.e. Hermitian), unbounded operator acting on a dense subspace of the quantum state space . If the operator acts on a ( normalizable ) quantum state then the operator is self-adjoint . In physics the term Hermitian often refers to both symmetric and self-adjoint operators. (In certain artificial situations, such as

4674-405: Is conserved under the evolution generated by B {\displaystyle B} . This implies a quantum version of the result proven by Emmy Noether in classical ( Lagrangian ) mechanics: for every differentiable symmetry of a Hamiltonian, there exists a corresponding conservation law . The simplest example of a quantum system with a position degree of freedom is a free particle in

4797-1066: Is considered as a sum over all possible classical and non-classical paths between the initial and final states. This is the quantum-mechanical counterpart of the action principle in classical mechanics. The Hamiltonian H {\displaystyle H} is known as the generator of time evolution, since it defines a unitary time-evolution operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} for each value of t {\displaystyle t} . From this relation between U ( t ) {\displaystyle U(t)} and H {\displaystyle H} , it follows that any observable A {\displaystyle A} that commutes with H {\displaystyle H} will be conserved : its expectation value will not change over time. This statement generalizes, as mathematically, any Hermitian operator A {\displaystyle A} can generate

4920-1007: Is denoted T ( ε ) , where ε represents the length of the translation. It satisfies the following identity: T ( ε ) | ψ ⟩ = ∫ d x T ( ε ) | x ⟩ ⟨ x | ψ ⟩ {\displaystyle T(\varepsilon )|\psi \rangle =\int dxT(\varepsilon )|x\rangle \langle x|\psi \rangle } that becomes ∫ d x | x + ε ⟩ ⟨ x | ψ ⟩ = ∫ d x | x ⟩ ⟨ x − ε | ψ ⟩ = ∫ d x | x ⟩ ψ ( x − ε ) {\displaystyle \int dx|x+\varepsilon \rangle \langle x|\psi \rangle =\int dx|x\rangle \langle x-\varepsilon |\psi \rangle =\int dx|x\rangle \psi (x-\varepsilon )} Assuming

5043-473: Is easily found to be the Gaussian ψ 0 ( x ) = C e − m ω x 2 2 ℏ . {\displaystyle \psi _{0}(x)=Ce^{-{\frac {m\omega x^{2}}{2\hbar }}}.} Conceptually, it is important that there is only one solution of this equation; if there were, say, two linearly independent ground states, we would get two independent chains of eigenvectors for

Quantum harmonic oscillator - Misplaced Pages Continue

5166-448: Is given by The operator U ( t ) = e − i H t / ℏ {\displaystyle U(t)=e^{-iHt/\hbar }} is known as the time-evolution operator, and has the crucial property that it is unitary . This time evolution is deterministic in the sense that – given an initial quantum state ψ ( 0 ) {\displaystyle \psi (0)} – it makes

5289-578: Is interpreted as momentum in the x -direction and E is the particle energy. The first order partial derivative with respect to space is ∂ ψ ( x , t ) ∂ x = i p ℏ e i ℏ ( p x − E t ) = i p ℏ ψ . {\displaystyle {\frac {\partial \psi (x,t)}{\partial x}}={\frac {ip}{\hbar }}e^{{\frac {i}{\hbar }}(px-Et)}={\frac {ip}{\hbar }}\psi .} This suggests

5412-406: Is its associated eigenvector. More generally, the eigenvalue is degenerate and the probability is given by ⟨ ψ , P λ ψ ⟩ {\displaystyle \langle \psi ,P_{\lambda }\psi \rangle } , where P λ {\displaystyle P_{\lambda }} is the projector onto its associated eigenspace. In

5535-436: Is its minimum value due to uncertainty relation and also corresponds to a gaussian wavefunction. This energy spectrum is noteworthy for three reasons. First, the energies are quantized, meaning that only discrete energy values (integer-plus-half multiples of ħω ) are possible; this is a general feature of quantum-mechanical systems when a particle is confined. Second, these discrete energy levels are equally spaced, unlike in

5658-726: Is known as wave–particle duality . In addition to light, electrons , atoms , and molecules are all found to exhibit the same dual behavior when fired towards a double slit. Another non-classical phenomenon predicted by quantum mechanics is quantum tunnelling : a particle that goes up against a potential barrier can cross it, even if its kinetic energy is smaller than the maximum of the potential. In classical mechanics this particle would be trapped. Quantum tunnelling has several important consequences, enabling radioactive decay , nuclear fusion in stars, and applications such as scanning tunnelling microscopy , tunnel diode and tunnel field-effect transistor . When quantum systems interact,

5781-558: Is known from classical mechanics , the momentum is the generator of translation , so the relation between translation and momentum operators is: T ( ε ) = 1 − i ℏ ε p ^ {\displaystyle T(\varepsilon )=1-{\frac {i}{\hbar }}\varepsilon {\hat {p}}} thus p ^ = − i ℏ d d x . {\displaystyle {\hat {p}}=-i\hbar {\frac {d}{dx}}.} Inserting

5904-584: Is known. The Hamiltonian of the particle is: H ^ = p ^ 2 2 m + 1 2 k x ^ 2 = p ^ 2 2 m + 1 2 m ω 2 x ^ 2 , {\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+{\frac {1}{2}}k{\hat {x}}^{2}={\frac {{\hat {p}}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}{\hat {x}}^{2}\,,} where m

6027-1590: Is not a simple shift in wavefunction phase. The time-evolved states are, however, also coherent states but with phase-shifting parameter α instead: α ( t ) = α ( 0 ) e − i ω t = α 0 e − i ω t {\displaystyle \alpha (t)=\alpha (0)e^{-i\omega t}=\alpha _{0}e^{-i\omega t}} . | α ( t ) ⟩ = ∑ n = 0 ∞ e − i ( n + 1 2 ) ω t | n ⟩ ⟨ n | α ⟩ = e − i ω t 2 e − 1 2 | α | 2 ∑ n = 0 ∞ ( α e − i ω t ) n n ! | n ⟩ = e − i ω t 2 | α e − i ω t ⟩ {\displaystyle |\alpha (t)\rangle =\sum _{n=0}^{\infty }e^{-i\left(n+{\frac {1}{2}}\right)\omega t}|n\rangle \langle n|\alpha \rangle =e^{\frac {-i\omega t}{2}}e^{-{\frac {1}{2}}|\alpha |^{2}}\sum _{n=0}^{\infty }{\frac {(\alpha e^{-i\omega t})^{n}}{\sqrt {n!}}}|n\rangle =e^{-{\frac {i\omega t}{2}}}|\alpha e^{-i\omega t}\rangle } Because

6150-444: Is not possible for the solution to be a single momentum eigenstate, or a single position eigenstate, as these are not normalizable quantum states. Instead, we can consider a Gaussian wave packet : which has Fourier transform, and therefore momentum distribution We see that as we make a {\displaystyle a} smaller the spread in position gets smaller, but the spread in momentum gets larger. Conversely, by making

6273-628: Is not sufficient for describing them at very small submicroscopic (atomic and subatomic ) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation, valid at large (macroscopic/microscopic) scale. Quantum systems have bound states that are quantized to discrete values of energy , momentum , angular momentum , and other quantities, in contrast to classical systems where these quantities can be measured continuously. Measurements of quantum systems show characteristics of both particles and waves ( wave–particle duality ), and there are limits to how accurately

SECTION 50

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6396-815: Is part of quantum communication protocols, such as quantum key distribution and superdense coding . Contrary to popular misconception, entanglement does not allow sending signals faster than light , as demonstrated by the no-communication theorem . Another possibility opened by entanglement is testing for " hidden variables ", hypothetical properties more fundamental than the quantities addressed in quantum theory itself, knowledge of which would allow more exact predictions than quantum theory provides. A collection of results, most significantly Bell's theorem , have demonstrated that broad classes of such hidden-variable theories are in fact incompatible with quantum physics. According to Bell's theorem, if nature actually operates in accord with any theory of local hidden variables, then

6519-535: Is postulated to be normalized under the Hilbert space inner product, that is, it obeys ⟨ ψ , ψ ⟩ = 1 {\displaystyle \langle \psi ,\psi \rangle =1} , and it is well-defined up to a complex number of modulus 1 (the global phase), that is, ψ {\displaystyle \psi } and e i α ψ {\displaystyle e^{i\alpha }\psi } represent

6642-466: Is replaced by − i ℏ ∂ ∂ x {\displaystyle -i\hbar {\frac {\partial }{\partial x}}} , and in particular in the non-relativistic Schrödinger equation in position space the momentum-squared term is replaced with a Laplacian times − ℏ 2 {\displaystyle -\hbar ^{2}} . When two different quantum systems are considered together,

6765-492: Is sometimes taken as one of the foundational postulates of quantum mechanics. The momentum and energy operators can be constructed in the following way. Starting in one dimension, using the plane wave solution to Schrödinger's equation of a single free particle, ψ ( x , t ) = e i ℏ ( p x − E t ) , {\displaystyle \psi (x,t)=e^{{\frac {i}{\hbar }}(px-Et)},} where p

6888-531: Is the 4-gradient , and the − iħ   becomes + iħ   preceding the 3-momentum operator. This operator occurs in relativistic quantum field theory , such as the Dirac equation and other relativistic wave equations , since energy and momentum combine into the 4-momentum vector above, momentum and energy operators correspond to space and time derivatives, and they need to be first order partial derivatives for Lorentz covariance . The Dirac operator and Dirac slash of

7011-796: Is the momentum operator (given by p ^ = − i ℏ ∂ / ∂ x {\displaystyle {\hat {p}}=-i\hbar \,\partial /\partial x} in the coordinate basis). The first term in the Hamiltonian represents the kinetic energy of the particle, and the second term represents its potential energy, as in Hooke's law . The time-independent Schrödinger equation (TISE) is, H ^ | ψ ⟩ = E | ψ ⟩   , {\displaystyle {\hat {H}}\left|\psi \right\rangle =E\left|\psi \right\rangle ~,} where E {\displaystyle E} denotes

7134-955: Is the unit operator . The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once. In quantum mechanics, position and momentum are conjugate variables . The following discussion uses the bra–ket notation . One may write ψ ( x ) = ⟨ x | ψ ⟩ = ∫ d p   ⟨ x | p ⟩ ⟨ p | ψ ⟩ = ∫ d p   e i x p / ℏ ψ ~ ( p ) 2 π ℏ , {\displaystyle \psi (x)=\langle x|\psi \rangle =\int \!\!dp~\langle x|p\rangle \langle p|\psi \rangle =\int \!\!dp~{e^{ixp/\hbar }{\tilde {\psi }}(p) \over {\sqrt {2\pi \hbar }}},} so

7257-497: Is the expression for the canonical momentum . For a charged particle q in an electromagnetic field , during a gauge transformation , the position space wave function undergoes a local U(1) group transformation, and p ^ ψ = − i ℏ ∂ ψ ∂ x {\textstyle {\hat {p}}\psi =-i\hbar {\frac {\partial \psi }{\partial x}}} will change its value. Therefore,

7380-422: Is the particle's mass, k is the force constant, ω = k / m {\textstyle \omega ={\sqrt {k/m}}} is the angular frequency of the oscillator, x ^ {\displaystyle {\hat {x}}} is the position operator (given by x in the coordinate basis), and p ^ {\displaystyle {\hat {p}}}

7503-415: Is the reduced Planck constant . The constant i ℏ {\displaystyle i\hbar } is introduced so that the Hamiltonian is reduced to the classical Hamiltonian in cases where the quantum system can be approximated by a classical system; the ability to make such an approximation in certain limits is called the correspondence principle . The solution of this differential equation

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7626-406: Is the same, except the gradient operator del is used instead of one partial derivative. In three dimensions, the plane wave solution to Schrödinger's equation is: ψ = e i ℏ ( p ⋅ r − E t ) {\displaystyle \psi =e^{{\frac {i}{\hbar }}(\mathbf {p} \cdot \mathbf {r} -Et)}} and the gradient

7749-469: Is then If the state for the first system is the vector ψ A {\displaystyle \psi _{A}} and the state for the second system is ψ B {\displaystyle \psi _{B}} , then the state of the composite system is Not all states in the joint Hilbert space H A B {\displaystyle {\mathcal {H}}_{AB}} can be written in this form, however, because

7872-445: The Bohr model of the atom, or the particle in a box . Third, the lowest achievable energy (the energy of the n = 0 state, called the ground state ) is not equal to the minimum of the potential well, but ħω /2 above it; this is called zero-point energy . Because of the zero-point energy, the position and momentum of the oscillator in the ground state are not fixed (as they would be in

7995-501: The Born rule : in the simplest case the eigenvalue λ {\displaystyle \lambda } is non-degenerate and the probability is given by | ⟨ λ → , ψ ⟩ | 2 {\displaystyle |\langle {\vec {\lambda }},\psi \rangle |^{2}} , where λ → {\displaystyle {\vec {\lambda }}}

8118-1225: The Hermite polynomials . To avoid confusion, these "natural units" will mostly not be adopted in this article. However, they frequently come in handy when performing calculations, by bypassing clutter. For example, the fundamental solution ( propagator ) of H − i∂ t , the time-dependent Schrödinger operator for this oscillator, simply boils down to the Mehler kernel , ⟨ x ∣ exp ⁡ ( − i t H ) ∣ y ⟩ ≡ K ( x , y ; t ) = 1 2 π i sin ⁡ t exp ⁡ ( i 2 sin ⁡ t ( ( x 2 + y 2 ) cos ⁡ t − 2 x y ) )   , {\displaystyle \langle x\mid \exp(-itH)\mid y\rangle \equiv K(x,y;t)={\frac {1}{\sqrt {2\pi i\sin t}}}\exp \left({\frac {i}{2\sin t}}\left((x^{2}+y^{2})\cos t-2xy\right)\right)~,} where K ( x , y ;0) = δ ( x − y ) . The most general solution for

8241-657: The canonical commutation relation , [ a , a † ] = 1 , [ N , a † ] = a † , [ N , a ] = − a , {\displaystyle [a,a^{\dagger }]=1,\qquad [N,a^{\dagger }]=a^{\dagger },\qquad [N,a]=-a,} and the Hamilton operator can be expressed as H ^ = ℏ ω ( N + 1 2 ) , {\displaystyle {\hat {H}}=\hbar \omega \left(N+{\frac {1}{2}}\right),} so

8364-713: The canonical commutation relation : Given a quantum state, the Born rule lets us compute expectation values for both X {\displaystyle X} and P {\displaystyle P} , and moreover for powers of them. Defining the uncertainty for an observable by a standard deviation , we have and likewise for the momentum: The uncertainty principle states that Either standard deviation can in principle be made arbitrarily small, but not both simultaneously. This inequality generalizes to arbitrary pairs of self-adjoint operators A {\displaystyle A} and B {\displaystyle B} . The commutator of these two operators

8487-650: The generators of normalized rotation in the phase space of x {\displaystyle x} and m d x d t {\displaystyle m{\frac {dx}{dt}}} , i.e they describe the forwards and backwards evolution in time of a classical harmonic oscillator. These operators lead to the following representation of x ^ {\displaystyle {\hat {x}}} and p ^ {\displaystyle {\hat {p}}} , x ^ = ℏ 2 m ω (

8610-423: The photoelectric effect . These early attempts to understand microscopic phenomena, now known as the " old quantum theory ", led to the full development of quantum mechanics in the mid-1920s by Niels Bohr , Erwin Schrödinger , Werner Heisenberg , Max Born , Paul Dirac and others. The modern theory is formulated in various specially developed mathematical formalisms . In one of them, a mathematical entity called

8733-404: The unit vectors for the three spatial dimensions, hence p ^ = − i ℏ ∇ {\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla } This momentum operator is in position space because the partial derivatives were taken with respect to the spatial variables. For a single particle with no electric charge and no spin ,

8856-562: The wave function provides information, in the form of probability amplitudes , about what measurements of a particle's energy, momentum, and other physical properties may yield. Quantum mechanics allows the calculation of properties and behaviour of physical systems. It is typically applied to microscopic systems: molecules, atoms and sub-atomic particles. It has been demonstrated to hold for complex molecules with thousands of atoms, but its application to human beings raises philosophical problems, such as Wigner's friend , and its application to

8979-941: The 3d momentum operator above and the energy operator into the 4-momentum (as a 1-form with (+ − − −) metric signature ): P μ = ( E c , − p ) {\displaystyle P_{\mu }=\left({\frac {E}{c}},-\mathbf {p} \right)} obtains the 4-momentum operator : P ^ μ = ( 1 c E ^ , − p ^ ) = i ℏ ( 1 c ∂ ∂ t , ∇ ) = i ℏ ∂ μ {\displaystyle {\hat {P}}_{\mu }=\left({\frac {1}{c}}{\hat {E}},-\mathbf {\hat {p}} \right)=i\hbar \left({\frac {1}{c}}{\frac {\partial }{\partial t}},\nabla \right)=i\hbar \partial _{\mu }} where ∂ μ

9102-475: The 4-momentum is given by contracting with the gamma matrices : γ μ P ^ μ = i ℏ γ μ ∂ μ = P ^ = i ℏ ∂ / {\displaystyle \gamma ^{\mu }{\hat {P}}_{\mu }=i\hbar \gamma ^{\mu }\partial _{\mu }={\hat {P}}=i\hbar \partial \!\!\!/} If

9225-431: The Hilbert space for the spin of a single proton is simply the space of two-dimensional complex vectors C 2 {\displaystyle \mathbb {C} ^{2}} with the usual inner product. Physical quantities of interest – position, momentum, energy, spin – are represented by observables, which are Hermitian (more precisely, self-adjoint ) linear operators acting on

9348-411: The Hilbert space of the combined system is the tensor product of the Hilbert spaces of the two components. For example, let A and B be two quantum systems, with Hilbert spaces H A {\displaystyle {\mathcal {H}}_{A}} and H B {\displaystyle {\mathcal {H}}_{B}} , respectively. The Hilbert space of the composite system

9471-432: The Hilbert space. A quantum state can be an eigenvector of an observable, in which case it is called an eigenstate , and the associated eigenvalue corresponds to the value of the observable in that eigenstate. More generally, a quantum state will be a linear combination of the eigenstates, known as a quantum superposition . When an observable is measured, the result will be one of its eigenvalues with probability given by

9594-489: The Schrödinger equation are known for very few relatively simple model Hamiltonians including the quantum harmonic oscillator , the particle in a box , the dihydrogen cation , and the hydrogen atom . Even the helium atom – which contains just two electrons – has defied all attempts at a fully analytic treatment, admitting no solution in closed form . However, there are techniques for finding approximate solutions. One method, called perturbation theory , uses

9717-425: The Schrödinger equation for the particle in a box are or, from Euler's formula , Momentum operator In a basis of Hilbert space consisting of momentum eigenstates expressed in the momentum representation, the action of the operator is simply multiplication by p , i.e. it is a multiplication operator , just as the position operator is a multiplication operator in the position representation. Note that

9840-483: The TISE for the Hamiltonian operator H ^ {\displaystyle {\hat {H}}} , is also one of its eigenstates with the corresponding eigenvalue: E n = ℏ ω ( n + 1 2 ) . {\displaystyle E_{n}=\hbar \omega \left(n+{\frac {1}{2}}\right).} QED. The commutation property yields N

9963-403: The analytic result for a simple quantum mechanical model to create a result for a related but more complicated model by (for example) the addition of a weak potential energy . Another approximation method applies to systems for which quantum mechanics produces only small deviations from classical behavior. These deviations can then be computed based on the classical motion. One consequence of

10086-606: The basic quantum formalism is the uncertainty principle. In its most familiar form, this states that no preparation of a quantum particle can imply simultaneously precise predictions both for a measurement of its position and for a measurement of its momentum. Both position and momentum are observables, meaning that they are represented by Hermitian operators . The position operator X ^ {\displaystyle {\hat {X}}} and momentum operator P ^ {\displaystyle {\hat {P}}} do not commute, but rather satisfy

10209-495: The canonical momentum is not gauge invariant , and hence not a measurable physical quantity. The kinetic momentum , a gauge invariant physical quantity, can be expressed in terms of the canonical momentum, the scalar potential   φ and vector potential   A : P ^ = − i ℏ ∇ − q A {\displaystyle \mathbf {\hat {P}} =-i\hbar \nabla -q\mathbf {A} } The expression above

10332-404: The collection of probability amplitudes that pertain to another. One consequence of the mathematical rules of quantum mechanics is a tradeoff in predictability between measurable quantities. The most famous form of this uncertainty principle says that no matter how a quantum particle is prepared or how carefully experiments upon it are arranged, it is impossible to have a precise prediction for

10455-626: The continuous case, these formulas give instead the probability density . After the measurement, if result λ {\displaystyle \lambda } was obtained, the quantum state is postulated to collapse to λ → {\displaystyle {\vec {\lambda }}} , in the non-degenerate case, or to P λ ψ / ⟨ ψ , P λ ψ ⟩ {\textstyle P_{\lambda }\psi {\big /}\!{\sqrt {\langle \psi ,P_{\lambda }\psi \rangle }}} , in

10578-485: The definition above is the canonical momentum , which is not gauge invariant and not a measurable physical quantity for charged particles in an electromagnetic field . In that case, the canonical momentum is not equal to the kinetic momentum . At the time quantum mechanics was developed in the 1920s, the momentum operator was found by many theoretical physicists, including Niels Bohr , Arnold Sommerfeld , Erwin Schrödinger , and Eugene Wigner . Its existence and form

10701-431: The dependence in position means that the momentum operator is equivalent (up to an i / ℏ {\displaystyle i/\hbar } factor) to taking the derivative according to the position, since in Fourier analysis differentiation corresponds to multiplication in the dual space . This is why in quantum equations in position space, the momentum p i {\displaystyle p_{i}}

10824-584: The eigenstates of N are also the eigenstates of energy. To see that, we can apply H ^ {\displaystyle {\hat {H}}} to a number state | n ⟩ {\displaystyle |n\rangle } : H ^ | n ⟩ = ℏ ω ( N ^ + 1 2 ) | n ⟩ . {\displaystyle {\hat {H}}|n\rangle =\hbar \omega \left({\hat {N}}+{\frac {1}{2}}\right)|n\rangle .} Using

10947-755: The energy eigenfunctions and eigenvalues simplify to Hermite functions and integers offset by a half, ψ n ( x ) = ⟨ x ∣ n ⟩ = 1 2 n n !   π − 1 / 4 exp ⁡ ( − x 2 / 2 )   H n ( x ) , {\displaystyle \psi _{n}(x)=\left\langle x\mid n\right\rangle ={1 \over {\sqrt {2^{n}n!}}}~\pi ^{-1/4}\exp(-x^{2}/2)~H_{n}(x),} E n = n + 1 2   , {\displaystyle E_{n}=n+{\tfrac {1}{2}}~,} where H n ( x ) are

11070-399: The energy spectrum given in the preceding section. Arbitrary eigenstates can be expressed in terms of |0⟩, | n ⟩ = ( a † ) n n ! | 0 ⟩ . {\displaystyle |n\rangle ={\frac {(a^{\dagger })^{n}}{\sqrt {n!}}}|0\rangle .} ⟨ n | a

11193-584: The figure; they are not eigenstates of the Hamiltonian. The " ladder operator " method, developed by Paul Dirac , allows extraction of the energy eigenvalues without directly solving the differential equation. It is generalizable to more complicated problems, notably in quantum field theory . Following this approach, we define the operators a and its adjoint a , a = m ω 2 ℏ ( x ^ + i m ω p ^ )

11316-876: The function ψ {\displaystyle \psi } to be analytic (i.e. differentiable in some domain of the complex plane ), one may expand in a Taylor series about x : ψ ( x − ε ) = ψ ( x ) − ε d ψ d x {\displaystyle \psi (x-\varepsilon )=\psi (x)-\varepsilon {\frac {d\psi }{dx}}} so for infinitesimal values of ε : T ( ε ) = 1 − ε d d x = 1 − i ℏ ε ( − i ℏ d d x ) {\displaystyle T(\varepsilon )=1-\varepsilon {d \over dx}=1-{i \over \hbar }\varepsilon \left(-i\hbar {d \over dx}\right)} As it

11439-415: The general case. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr–Einstein debates , in which the two scientists attempted to clarify these fundamental principles by way of thought experiments . In the decades after the formulation of quantum mechanics,

11562-504: The harmonic oscillator. Once the ground state is computed, one can show inductively that the excited states are Hermite polynomials times the Gaussian ground state, using the explicit form of the raising operator in the position representation. One can also prove that, as expected from the uniqueness of the ground state, the Hermite functions energy eigenstates ψ n {\displaystyle \psi _{n}} constructed by

11685-462: The interference pattern appears via the varying density of these particle hits on the screen. Furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit (as would a classical particle), and not through both slits (as would a wave). However, such experiments demonstrate that particles do not form the interference pattern if one detects which slit they pass through. This behavior

11808-1331: The ladder method form a complete orthonormal set of functions. Explicitly connecting with the previous section, the ground state |0⟩ in the position representation is determined by a | 0 ⟩ = 0 {\displaystyle a|0\rangle =0} , ⟨ x ∣ a ∣ 0 ⟩ = 0 ⇒ ( x + ℏ m ω d d x ) ⟨ x ∣ 0 ⟩ = 0 ⇒ {\displaystyle \left\langle x\mid a\mid 0\right\rangle =0\qquad \Rightarrow \left(x+{\frac {\hbar }{m\omega }}{\frac {d}{dx}}\right)\left\langle x\mid 0\right\rangle =0\qquad \Rightarrow } ⟨ x ∣ 0 ⟩ = ( m ω π ℏ ) 1 4 exp ⁡ ( − m ω 2 ℏ x 2 ) = ψ 0   , {\displaystyle \left\langle x\mid 0\right\rangle =\left({\frac {m\omega }{\pi \hbar }}\right)^{\frac {1}{4}}\exp \left(-{\frac {m\omega }{2\hbar }}x^{2}\right)=\psi _{0}~,} hence ⟨ x ∣

11931-430: The light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere , producing bright and dark bands on the screen – a result that would not be expected if light consisted of classical particles. However, the light is always found to be absorbed at the screen at discrete points, as individual particles rather than waves;

12054-402: The momentum acting in coordinate space corresponds to spatial frequency, ⟨ x | p ^ | ψ ⟩ = − i ℏ d d x ψ ( x ) . {\displaystyle \langle x|{\hat {p}}|\psi \rangle =-i\hbar {\frac {d}{dx}}\psi (x).} An analogous result applies for

12177-642: The momentum operator can be written in the position basis as: p ^ = − i ℏ ∇ {\displaystyle \mathbf {\hat {p}} =-i\hbar \nabla } where ∇ is the gradient operator, ħ is the reduced Planck constant , and i is the imaginary unit . In one spatial dimension, this becomes p ^ = p ^ x = − i ℏ ∂ ∂ x . {\displaystyle {\hat {p}}={\hat {p}}_{x}=-i\hbar {\partial \over \partial x}.} This

12300-432: The momentum operator with momentum p = ℏ k {\displaystyle p=\hbar k} . The coefficients of the superposition are ψ ^ ( k , 0 ) {\displaystyle {\hat {\psi }}(k,0)} , which is the Fourier transform of the initial quantum state ψ ( x , 0 ) {\displaystyle \psi (x,0)} . It

12423-413: The oldest and most common is the " transformation theory " proposed by Paul Dirac , which unifies and generalizes the two earliest formulations of quantum mechanics – matrix mechanics (invented by Werner Heisenberg ) and wave mechanics (invented by Erwin Schrödinger ). An alternative formulation of quantum mechanics is Feynman 's path integral formulation , in which a quantum-mechanical amplitude

12546-412: The one-dimensional case in the x {\displaystyle x} direction, the time-independent Schrödinger equation may be written With the differential operator defined by with state ψ {\displaystyle \psi } in this case having energy E {\displaystyle E} coincident with the kinetic energy of the particle. The general solutions of

12669-415: The operator equivalence p ^ = − i ℏ ∂ ∂ x {\displaystyle {\hat {p}}=-i\hbar {\frac {\partial }{\partial x}}} so the momentum of the particle and the value that is measured when a particle is in a plane wave state is the (generalized) eigenvalue of the above operator. Since the partial derivative

12792-449: The original quantum system ceases to exist as an independent entity (see Measurement in quantum mechanics ). The time evolution of a quantum state is described by the Schrödinger equation: Here H {\displaystyle H} denotes the Hamiltonian , the observable corresponding to the total energy of the system, and ℏ {\displaystyle \hbar }

12915-738: The physicists' Hermite polynomials , H n ( z ) = ( − 1 ) n   e z 2 d n d z n ( e − z 2 ) . {\displaystyle H_{n}(z)=(-1)^{n}~e^{z^{2}}{\frac {d^{n}}{dz^{n}}}\left(e^{-z^{2}}\right).} The corresponding energy levels are E n = ℏ ω ( n + 1 2 ) . {\displaystyle E_{n}=\hbar \omega {\bigl (}n+{\tfrac {1}{2}}{\bigr )}.} The expectation values of position and momentum combined with variance of each variable can be derived from

13038-428: The position becomes more and more uncertain. The uncertainty in momentum, however, stays constant. The particle in a one-dimensional potential energy box is the most mathematically simple example where restraints lead to the quantization of energy levels. The box is defined as having zero potential energy everywhere inside a certain region, and therefore infinite potential energy everywhere outside that region. For

13161-1099: The position operator in the momentum basis, ⟨ p | x ^ | ψ ⟩ = i ℏ d d p ψ ( p ) , {\displaystyle \langle p|{\hat {x}}|\psi \rangle =i\hbar {\frac {d}{dp}}\psi (p),} leading to further useful relations, ⟨ p | x ^ | p ′ ⟩ = i ℏ d d p δ ( p − p ′ ) , {\displaystyle \langle p|{\hat {x}}|p'\rangle =i\hbar {\frac {d}{dp}}\delta (p-p'),} ⟨ x | p ^ | x ′ ⟩ = − i ℏ d d x δ ( x − x ′ ) , {\displaystyle \langle x|{\hat {p}}|x'\rangle =-i\hbar {\frac {d}{dx}}\delta (x-x'),} where δ stands for Dirac's delta function . The translation operator

13284-456: The problem. These can be found by nondimensionalization . The result is that, if energy is measured in units of ħω and distance in units of √ ħ /( mω ) , then the Hamiltonian simplifies to H = − 1 2 d 2 d x 2 + 1 2 x 2 , {\displaystyle H=-{\frac {1}{2}}{d^{2} \over dx^{2}}+{\frac {1}{2}}x^{2},} while

13407-680: The property of the number operator N ^ {\displaystyle {\hat {N}}} : N ^ | n ⟩ = n | n ⟩ , {\displaystyle {\hat {N}}|n\rangle =n|n\rangle ,} we get: H ^ | n ⟩ = ℏ ω ( n + 1 2 ) | n ⟩ . {\displaystyle {\hat {H}}|n\rangle =\hbar \omega \left(n+{\frac {1}{2}}\right)|n\rangle .} Thus, since | n ⟩ {\displaystyle |n\rangle } solves

13530-880: The quantum states on the semi-infinite interval [0, ∞) , there is no way to make the momentum operator Hermitian. This is closely related to the fact that a semi-infinite interval cannot have translational symmetry—more specifically, it does not have unitary translation operators . See below .) By applying the commutator to an arbitrary state in either the position or momentum basis, one can easily show that: [ x ^ , p ^ ] = x ^ p ^ − p ^ x ^ = i ℏ I , {\displaystyle \left[{\hat {x}},{\hat {p}}\right]={\hat {x}}{\hat {p}}-{\hat {p}}{\hat {x}}=i\hbar \mathbb {I} ,} where I {\displaystyle \mathbb {I} }

13653-400: The question of what constitutes a "measurement" has been extensively studied. Newer interpretations of quantum mechanics have been formulated that do away with the concept of " wave function collapse " (see, for example, the many-worlds interpretation ). The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wave functions become entangled so that

13776-788: The raising operator and multiplying by suitable normalization factors , we can produce an infinite set of energy eigenstates { | 0 ⟩ , | 1 ⟩ , | 2 ⟩ , … , | n ⟩ , … } , {\displaystyle \left\{\left|0\right\rangle ,\left|1\right\rangle ,\left|2\right\rangle ,\ldots ,\left|n\right\rangle ,\ldots \right\},} such that H ^ | n ⟩ = ℏ ω ( n + 1 2 ) | n ⟩ , {\displaystyle {\hat {H}}\left|n\right\rangle =\hbar \omega \left(n+{\frac {1}{2}}\right)\left|n\right\rangle ,} which matches

13899-460: The relations above, we can also define a number operator N , which has the following property: N = a † a N | n ⟩ = n | n ⟩ . {\displaystyle {\begin{aligned}N&=a^{\dagger }a\\N\left|n\right\rangle &=n\left|n\right\rangle .\end{aligned}}} The following commutators can be easily obtained by substituting

14022-413: The result can be the creation of quantum entanglement : their properties become so intertwined that a description of the whole solely in terms of the individual parts is no longer possible. Erwin Schrödinger called entanglement "... the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought". Quantum entanglement enables quantum computing and

14145-566: The results of a Bell test will be constrained in a particular, quantifiable way. Many Bell tests have been performed and they have shown results incompatible with the constraints imposed by local hidden variables. It is not possible to present these concepts in more than a superficial way without introducing the mathematics involved; understanding quantum mechanics requires not only manipulating complex numbers, but also linear algebra , differential equations , group theory , and other more advanced subjects. Accordingly, this article will present

14268-463: The same physical system. In other words, the possible states are points in the projective space of a Hilbert space, usually called the complex projective space . The exact nature of this Hilbert space is dependent on the system – for example, for describing position and momentum the Hilbert space is the space of complex square-integrable functions L 2 ( C ) {\displaystyle L^{2}(\mathbb {C} )} , while

14391-529: The state's energy coincides with the potential energy. (See the discussion below of the highly excited states.) This is consistent with the classical harmonic oscillator, in which the particle spends more of its time (and is therefore more likely to be found) near the turning points, where it is moving the slowest. The correspondence principle is thus satisfied. Moreover, special nondispersive wave packets , with minimum uncertainty, called coherent states oscillate very much like classical objects, as illustrated in

14514-625: The superposition principle implies that linear combinations of these "separable" or "product states" are also valid. For example, if ψ A {\displaystyle \psi _{A}} and ϕ A {\displaystyle \phi _{A}} are both possible states for system A {\displaystyle A} , and likewise ψ B {\displaystyle \psi _{B}} and ϕ B {\displaystyle \phi _{B}} are both possible states for system B {\displaystyle B} , then

14637-441: The theory is that it usually cannot predict with certainty what will happen, but only give probabilities. Mathematically, a probability is found by taking the square of the absolute value of a complex number , known as a probability amplitude. This is known as the Born rule , named after physicist Max Born . For example, a quantum particle like an electron can be described by a wave function, which associates to each point in space

14760-569: The tilde represents the Fourier transform, in converting from coordinate space to momentum space. It then holds that p ^ = ∫ d p   | p ⟩ p ⟨ p | = − i ℏ ∫ d x   | x ⟩ d d x ⟨ x |   , {\displaystyle {\hat {p}}=\int \!\!dp~|p\rangle p\langle p|=-i\hbar \int \!\!dx~|x\rangle {\frac {d}{dx}}\langle x|~,} that is,

14883-437: The universe as a whole remains speculative. Predictions of quantum mechanics have been verified experimentally to an extremely high degree of accuracy . For example, the refinement of quantum mechanics for the interaction of light and matter, known as quantum electrodynamics (QED), has been shown to agree with experiment to within 1 part in 10 when predicting the magnetic properties of an electron. A fundamental feature of

15006-519: The value of a physical quantity can be predicted prior to its measurement, given a complete set of initial conditions (the uncertainty principle ). Quantum mechanics arose gradually from theories to explain observations that could not be reconciled with classical physics, such as Max Planck 's solution in 1900 to the black-body radiation problem, and the correspondence between energy and frequency in Albert Einstein 's 1905 paper , which explained

15129-1331: The wavefunction to understand the behavior of the energy eigenkets. They are shown to be ⟨ x ^ ⟩ = 0 {\textstyle \langle {\hat {x}}\rangle =0} and ⟨ p ^ ⟩ = 0 {\textstyle \langle {\hat {p}}\rangle =0} owing to the symmetry of the problem, whereas: ⟨ x ^ 2 ⟩ = ( 2 n + 1 ) ℏ 2 m ω = σ x 2 {\displaystyle \langle {\hat {x}}^{2}\rangle =(2n+1){\frac {\hbar }{2m\omega }}=\sigma _{x}^{2}} ⟨ p ^ 2 ⟩ = ( 2 n + 1 ) m ℏ ω 2 = σ p 2 {\displaystyle \langle {\hat {p}}^{2}\rangle =(2n+1){\frac {m\hbar \omega }{2}}=\sigma _{p}^{2}} The variance in both position and momentum are observed to increase for higher energy levels. The lowest energy level has value of σ x σ p = ℏ 2 {\textstyle \sigma _{x}\sigma _{p}={\frac {\hbar }{2}}} which

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