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105-429: In some contrast to the wave formulation, it produces spectra of (mostly energy) operators by purely algebraic, ladder operator methods. Relying on these methods, Wolfgang Pauli derived the hydrogen atom spectrum in 1926, before the development of wave mechanics. In 1925, Werner Heisenberg , Max Born , and Pascual Jordan formulated the matrix mechanics representation of quantum mechanics. In 1925 Werner Heisenberg

210-539: A ^ † = m ω 2 ℏ ( x ^ − i m ω p ^ ) . {\displaystyle {\begin{aligned}{\hat {a}}&={\sqrt {m\omega \over 2\hbar }}\left({\hat {x}}+{i \over m\omega }{\hat {p}}\right),\\{\hat {a}}^{\dagger }&={\sqrt {m\omega \over 2\hbar }}\left({\hat {x}}-{i \over m\omega }{\hat {p}}\right).\end{aligned}}} They provide

315-460: A raising or lowering operator (collectively known as ladder operators ) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator , and the lowering operator the annihilation operator . Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum . There

420-505: A "bad conscience" that he alone had received the Prize "for work done in Göttingen in collaboration – you, Jordan and I". Heisenberg went on to say that Born and Jordan's contribution to quantum mechanics cannot be changed by "a wrong decision from the outside". In 1954, Heisenberg wrote an article honoring Max Planck for his insight in 1900. In the article, Heisenberg credited Born and Jordan for

525-595: A convenient means to extract energy eigenvalues without directly solving the system's differential equation. There are two main approaches given in the literature using ladder operators, one using the Laplace–Runge–Lenz vector, another using factorization of the Hamiltonian. Another application of the ladder operator concept is found in the quantum mechanical treatment of the electronic energy of hydrogen-like atoms and ions. The Laplace–Runge–Lenz vector commutes with

630-503: A definite position and momentum . This principle of uncertainty holds for many other pairs of observables as well. For example, the energy does not commute with the position either, so it is impossible to precisely determine the position and energy of an electron in an atom. In 1928, Albert Einstein nominated Heisenberg, Born, and Jordan for the Nobel Prize in Physics . The announcement of

735-457: A general angular momentum vector J with components J x , J y and J z one defines the two ladder operators J + = J x + i J y , J − = J x − i J y , {\displaystyle {\begin{aligned}J_{+}&=J_{x}+iJ_{y},\\J_{-}&=J_{x}-iJ_{y},\end{aligned}}} where i

840-477: A great many of the mathematical tools necessary for the continued development of quantum mechanics. In 1926, John von Neumann became assistant to David Hilbert, and he would coin the term Hilbert space to describe the algebra and analysis which were used in the development of quantum mechanics. A linchpin contribution to this formulation was achieved in Dirac's reinterpretation/synthesis paper of 1925, which invented

945-506: A homogeneous dielectric sphere, which was published in 1908 under the title of "Contributions to the optics of turbid media, particularly of colloidal metal solutions" in Annalen der Physik . The term Mie scattering is still related to his name. Using Maxwell's electromagnetic theory applied to spherical gold particles Mie provided a theoretical treatment of plasmon resonance absorption of gold colloids . The sharp absorption bands depend on

1050-580: A non-negative half -integer multiple of ħ . Gustav Mie Gustav Adolf Feodor Wilhelm Ludwig Mie ( German: [miː] ; 29 September 1868 – 13 February 1957) was a German physicist . His work included Mie scattering , Mie potential , the Mie–Grüneisen equation of state and an early effort at classical unified field theories . Mie was born in Rostock , Mecklenburg-Schwerin , Germany in 1868. From 1886 he studied mathematics and physics at

1155-484: A photon, then ended up in orbit number m , the energy of the photon is E n − E m , which means that its frequency is ( E n − E m )/ h . For large n and m , but with n − m relatively small, these are the classical frequencies by Bohr 's correspondence principle E n − E m ≈ h ( n − m ) / T . {\displaystyle E_{n}-E_{m}\approx h(n-m)/T.} In

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1260-462: A positive integer multiple of the Planck constant ∫ 0 T P d X d t d t = ∫ 0 T P d X = n h . {\displaystyle \int _{0}^{T}P\;{dX \over dt}\;dt=\int _{0}^{T}P\;dX=nh.} While this restriction correctly selects orbits with more or less the right energy values E n ,

1365-463: A representation theory standpoint a linear representation of a semi-simple Lie group in continuous real parameters induces a set of generators for the Lie algebra . A complex linear combination of those are the ladder operators. For each parameter there is a set of ladder operators; these are then a standardized way to navigate one dimension of the root system and root lattice . The ladder operators of

1470-404: A single frequency, and X and P can be recovered from their sum and difference. Since A ( t ) has a classical Fourier series with only the lowest frequency, and the matrix element A mn is the ( m − n ) th Fourier coefficient of the classical orbit, the matrix for A is nonzero only on the line just above the diagonal, where it is equal to √ 2 E n . The matrix for A

1575-551: A suitable equation for C l {\displaystyle C_{l}} is C l = p r + i ℏ ( l + 1 ) r − i B l + 1 {\displaystyle C_{l}=p_{r}+{\frac {i\hbar (l+1)}{r}}-{\frac {iB}{l+1}}} with F l = G l = B 2 ( l + 1 ) 2 . {\displaystyle F_{l}=G_{l}={\frac {B^{2}}{(l+1)^{2}}}.} There

1680-473: A total degeneracy of ( n + 1 ) ( n + 2 ) / 2 {\displaystyle (n+1)(n+2)/2} The degeneracies of the 3D isotropic harmonic oscillator are related to the special unitary group SU(3) Many sources credit Paul Dirac with the invention of ladder operators. Dirac's use of the ladder operators shows that the total angular momentum quantum number j {\displaystyle j} needs to be

1785-476: Is iħ , multiplied by the identity . It is likewise simple to verify that the matrix H = 1 2 ( X 2 + P 2 ) {\displaystyle H={1 \over 2}(X^{2}+P^{2})} is a diagonal matrix , with eigenvalues E i . The harmonic oscillator is an important case. Finding the matrices is easier than determining the general conditions from these special forms. For this reason, Heisenberg investigated

1890-402: Is a relationship between the raising and lowering ladder operators and the creation and annihilation operators commonly used in quantum field theory which lies in representation theory . The creation operator a i increments the number of particles in state i , while the corresponding annihilation operator a i decrements the number of particles in state i . This clearly satisfies

1995-566: Is an eigenstate of H l + 1 {\displaystyle H_{l+1}} with eigenvalue E l + 1 n ′ = E l n + ( F l − G l ) / ( 2 μ ) . {\displaystyle E_{l+1}^{n'}=E_{l}^{n}+(F_{l}-G_{l})/(2\mu ).} If F l = G l {\displaystyle F_{l}=G_{l}} , then n ′ = n {\displaystyle n'=n} , and

2100-486: Is an eigenstate of N with eigenvalue n , then X | n ⟩ {\displaystyle X|n\rangle } is an eigenstate of N with eigenvalue n + c or is zero. The operator X is a raising operator for N if c is real and positive, and a lowering operator for N if c is real and negative. If N is a Hermitian operator , then c must be real, and the Hermitian adjoint of X obeys

2205-1067: Is an eigenstate of N with eigenvalue equation N | n ⟩ = n | n ⟩ , {\displaystyle N|n\rangle =n|n\rangle ,} then the operator X acts on | n ⟩ {\displaystyle |n\rangle } in such a way as to shift the eigenvalue by c : N X | n ⟩ = ( X N + [ N , X ] ) | n ⟩ = X N | n ⟩ + [ N , X ] | n ⟩ = X n | n ⟩ + c X | n ⟩ = ( n + c ) X | n ⟩ . {\displaystyle {\begin{aligned}NX|n\rangle &=(XN+[N,X])|n\rangle \\&=XN|n\rangle +[N,X]|n\rangle \\&=Xn|n\rangle +cX|n\rangle \\&=(n+c)X|n\rangle .\end{aligned}}} In other words, if | n ⟩ {\displaystyle |n\rangle }

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2310-1078: Is an upper bound to the ladder operator if the energy is negative (so C l | n l max ⟩ = 0 {\displaystyle C_{l}|nl_{\text{max}}\rangle =0} for some l max {\displaystyle l_{\text{max}}} ), then if follows from equation ( 1 ) that E l n = − F l / 2 μ = − B 2 2 μ ( l max + 1 ) 2 = − μ Z 2 e 4 2 ℏ 2 ( l max + 1 ) 2 , {\displaystyle E_{l}^{n}=-F_{l}/{2\mu }=-{\frac {B^{2}}{2\mu (l_{\text{max}}+1)^{2}}}=-{\frac {\mu Z^{2}e^{4}}{2\hbar ^{2}(l_{\text{max}}+1)^{2}}},} and n {\displaystyle n} can be identified with l max + 1. {\displaystyle l_{\text{max}}+1.} Whenever there

2415-508: Is bounded by the value of j ( − j ≤ m ≤ j {\displaystyle -j\leq m\leq j} ), one has J + | j , + j ⟩ = 0 , J − | j , − j ⟩ = 0. {\displaystyle {\begin{aligned}J_{+}|j,\,+j\rangle &=0,\\J_{-}|j,\,-j\rangle &=0.\end{aligned}}} The above demonstration

2520-646: Is decreasing in energy by ω ℏ {\displaystyle \omega \hbar } unless C l | n , l ⟩ = 0 {\displaystyle C_{l}|n,l\rangle =0} for some value of l {\displaystyle l} . Identifying this value as n {\displaystyle n} gives E l n = − F l = ( n + 3 2 ) ω ℏ . {\displaystyle E_{l}^{n}=-F_{l}=\left(n+{\tfrac {3}{2}}\right)\omega \hbar .} It then follows

2625-456: Is degeneracy caused from angular momentum; there is additional degeneracy caused by the oscillator potential. Consider the states | n , n ⟩ , | n − 1 , n − 1 ⟩ , | n − 2 , n − 2 ⟩ , … {\displaystyle |n,\,n\rangle ,|n-1,\,n-1\rangle ,|n-2,\,n-2\rangle ,\dots } and apply

2730-647: Is degeneracy in a system, there is usually a related symmetry property and group. The degeneracy of the energy levels for the same value of n {\displaystyle n} but different angular momenta has been identified as the SO(4) symmetry of the spherically symmetric Coulomb potential. The 3D isotropic harmonic oscillator has a potential given by V ( r ) = 1 2 μ ω 2 r 2 . {\displaystyle V(r)={\tfrac {1}{2}}\mu \omega ^{2}r^{2}.} It can similarly be managed using

2835-601: Is effectively the construction of the Clebsch–Gordan coefficients . Many terms in the Hamiltonians of atomic or molecular systems involve the scalar product of angular momentum operators. An example is the magnetic dipole term in the hyperfine Hamiltonian : H ^ D = A ^ I ⋅ J , {\displaystyle {\hat {H}}_{\text{D}}={\hat {A}}\mathbf {I} \cdot \mathbf {J} ,} where I

2940-2071: Is likewise only nonzero on the line below the diagonal, with the same elements. Thus, from A and A , reconstruction yields 2 X ( 0 ) = ℏ [ 0 1 0 0 0 ⋯ 1 0 2 0 0 ⋯ 0 2 0 3 0 ⋯ 0 0 3 0 4 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ] , {\displaystyle {\sqrt {2}}X(0)={\sqrt {\hbar }}\;{\begin{bmatrix}0&{\sqrt {1}}&0&0&0&\cdots \\{\sqrt {1}}&0&{\sqrt {2}}&0&0&\cdots \\0&{\sqrt {2}}&0&{\sqrt {3}}&0&\cdots \\0&0&{\sqrt {3}}&0&{\sqrt {4}}&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \\\end{bmatrix}},} and 2 P ( 0 ) = ℏ [ 0 − i 1 0 0 0 ⋯ i 1 0 − i 2 0 0 ⋯ 0 i 2 0 − i 3 0 ⋯ 0 0 i 3 0 − i 4 ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ] , {\displaystyle {\sqrt {2}}P(0)={\sqrt {\hbar }}\;{\begin{bmatrix}0&-i{\sqrt {1}}&0&0&0&\cdots \\i{\sqrt {1}}&0&-i{\sqrt {2}}&0&0&\cdots \\0&i{\sqrt {2}}&0&-i{\sqrt {3}}&0&\cdots \\0&0&i{\sqrt {3}}&0&-i{\sqrt {4}}&\cdots \\\vdots &\vdots &\vdots &\vdots &\vdots &\ddots \\\end{bmatrix}},} which, up to

3045-404: Is not hermitian, but has a real and imaginary part. The real part is one half the symmetric expression XP + PX , while the imaginary part is proportional to the commutator [ X , P ] = ( X P − P X ) . {\displaystyle [X,P]=(XP-PX).} It is simple to verify explicitly that XP − PX in the case of the harmonic oscillator,

3150-660: Is some scalar multiplied by | j ( m ± 1 ) ⟩ {\displaystyle {|j\,(m\pm 1)\rangle }} : J + | j m ⟩ = α | j ( m + 1 ) ⟩ , J − | j m ⟩ = β | j ( m − 1 ) ⟩ . {\displaystyle {\begin{aligned}J_{+}|j\,m\rangle &=\alpha |j\,(m+1)\rangle ,\\J_{-}|j\,m\rangle &=\beta |j\,(m-1)\rangle .\end{aligned}}} This illustrates

3255-405: Is still vague and unclear to me, but it seems as if the electrons will no more move on orbits. On July 9 Heisenberg gave the same paper of his calculations to Max Born, saying that "he had written a crazy paper and did not dare to send it in for publication, and that Born should read it and advise him" prior to publication. Heisenberg then departed for a while, leaving Born to analyse the paper. In

Matrix mechanics - Misplaced Pages Continue

3360-589: Is that it clearly indicates which states are coupled by this term in the Hamiltonian, that is those with quantum numbers differing by m i = ±1 and m j = ∓1 only . Another application of the ladder operator concept is found in the quantum-mechanical treatment of the harmonic oscillator. We can define the lowering and raising operators as a ^ = m ω 2 ℏ ( x ^ + i m ω p ^ ) ,

3465-493: Is the imaginary unit . The commutation relation between the cartesian components of any angular momentum operator is given by [ J i , J j ] = i ℏ ϵ i j k J k , {\displaystyle [J_{i},J_{j}]=i\hbar \epsilon _{ijk}J_{k},} where ε ijk is the Levi-Civita symbol , and each of i , j and k can take any of

3570-1629: Is the Lie algebra of s l ( 2 , R ) {\displaystyle {{\mathfrak {s}}l}(2,\mathbb {R} )} ). The properties of the ladder operators can be determined by observing how they modify the action of the J z operator on a given state: J z J ± | j m ⟩ = ( J ± J z + [ J z , J ± ] ) | j m ⟩ = ( J ± J z ± ℏ J ± ) | j m ⟩ = ℏ ( m ± 1 ) J ± | j m ⟩ . {\displaystyle {\begin{aligned}J_{z}J_{\pm }|j\,m\rangle &={\big (}J_{\pm }J_{z}+[J_{z},J_{\pm }]{\big )}|j\,m\rangle \\&=(J_{\pm }J_{z}\pm \hbar J_{\pm })|j\,m\rangle \\&=\hbar (m\pm 1)J_{\pm }|j\,m\rangle .\end{aligned}}} Compare this result with J z | j ( m ± 1 ) ⟩ = ℏ ( m ± 1 ) | j ( m ± 1 ) ⟩ . {\displaystyle J_{z}|j\,(m\pm 1)\rangle =\hbar (m\pm 1)|j\,(m\pm 1)\rangle .} Thus, one concludes that J ± | j m ⟩ {\displaystyle {J_{\pm }|j\,m\rangle }}

3675-2526: Is the angular momentum, p → {\displaystyle {\vec {p}}} is the linear momentum, μ {\displaystyle \mu } is the reduced mass of the system, e {\displaystyle e} is the electronic charge, and Z {\displaystyle Z} is the atomic number of the nucleus. Analogous to the angular momentum ladder operators, one has A + = A x + i A y {\displaystyle A_{+}=A_{x}+iA_{y}} and A − = A x − i A y {\displaystyle A_{-}=A_{x}-iA_{y}} . The commutators needed to proceed are [ A ± , L z ] = ∓ i ℏ A ∓ {\displaystyle [A_{\pm },L_{z}]=\mp {\boldsymbol {i}}\hbar A_{\mp }} and [ A ± , L 2 ] = ∓ 2 ℏ 2 A ± − 2 ℏ A ± L z ± 2 ℏ A z L ± . {\displaystyle [A_{\pm },L^{2}]=\mp 2\hbar ^{2}A_{\pm }-2\hbar A_{\pm }L_{z}\pm 2\hbar A_{z}L_{\pm }.} Therefore, A + | ? , ℓ , m ℓ ⟩ → | ? , ℓ , m ℓ + 1 ⟩ {\displaystyle A_{+}|?,\ell ,m_{\ell }\rangle \rightarrow |?,\ell ,m_{\ell }+1\rangle } and − L 2 ( A + | ? , ℓ , ℓ ⟩ ) = − ℏ 2 ( ℓ + 1 ) ( ( ℓ + 1 ) + 1 ) ( A + | ? , ℓ , ℓ ⟩ ) , {\displaystyle -L^{2}\left(A_{+}|?,\ell ,\ell \rangle \right)=-\hbar ^{2}(\ell +1)((\ell +1)+1)\left(A_{+}|?,\ell ,\ell \rangle \right),} so A + | ? , ℓ , ℓ ⟩ → | ? , ℓ + 1 , ℓ + 1 ⟩ , {\displaystyle A_{+}|?,\ell ,\ell \rangle \rightarrow |?,\ell +1,\ell +1\rangle ,} where

3780-1061: Is the nuclear spin. The angular momentum algebra can often be simplified by recasting it in the spherical basis . Using the notation of spherical tensor operators , the "−1", "0" and "+1" components of J ≡ J are given by J − 1 ( 1 ) = 1 2 ( J x − i J y ) = J − 2 , J 0 ( 1 ) = J z , J + 1 ( 1 ) = − 1 2 ( J x + i J y ) = − J + 2 . {\displaystyle {\begin{aligned}J_{-1}^{(1)}&={\dfrac {1}{\sqrt {2}}}(J_{x}-iJ_{y})={\dfrac {J_{-}}{\sqrt {2}}},\\J_{0}^{(1)}&=J_{z},\\J_{+1}^{(1)}&=-{\frac {1}{\sqrt {2}}}(J_{x}+iJ_{y})=-{\frac {J_{+}}{\sqrt {2}}}.\end{aligned}}} From these definitions, it can be shown that

3885-461: Is the original form of Heisenberg's equation of motion. Given two arrays X nm and P nm describing two physical quantities, Heisenberg could form a new array of the same type by combining the terms X nk P km , which also oscillate with the right frequency. Since the Fourier coefficients of the product of two quantities is the convolution of the Fourier coefficients of each one separately,

3990-550: Is the traditional quantum number. The Hamiltonian for a hydrogen-like potential can be written in spherical coordinates as H = 1 2 μ [ p r 2 + 1 r 2 L 2 ] + V ( r ) , {\displaystyle H={\frac {1}{2\mu }}\left[p_{r}^{2}+{\frac {1}{r^{2}}}L^{2}\right]+V(r),} where V ( r ) = − Z e 2 / r {\displaystyle V(r)=-Ze^{2}/r} , and

4095-707: The n ′ = n − 1 {\displaystyle n'=n-1} so that C l | n l ⟩ = λ l n | n − 1 , l + 1 ⟩ , {\displaystyle C_{l}|nl\rangle =\lambda _{l}^{n}|n-1,\,l+1\rangle ,} giving a recursion relation on λ {\displaystyle \lambda } with solution λ l n = − μ ω ℏ 2 ( n − l ) . {\displaystyle \lambda _{l}^{n}=-\mu \omega \hbar {\sqrt {2(n-l)}}.} There

4200-553: The Rydberg formula E n = − μ Z 2 e 4 2 ℏ 2 ( ℓ ∗ + 1 ) 2 , {\displaystyle E_{n}=-{\frac {\mu Z^{2}e^{4}}{2\hbar ^{2}(\ell ^{*}+1)^{2}}},} implying that ℓ ∗ + 1 = n = ? {\displaystyle \ell ^{*}+1=n=?} , where n {\displaystyle n}

4305-536: The University of Freiburg , where he worked up to his retirement in 1935. In Freiburg, during the Nazi dictatorship, Mie was member of the university opposition of the so-called "Freiburger Kreis" ( Freiburg Circles ) and one of the participants of the original "Freiburger Konzil". He died at Freiburg im Breisgau in 1957. During his Greifswald years Mie worked on the computation of scattering of an electromagnetic wave by

Matrix mechanics - Misplaced Pages Continue

4410-449: The University of Rostock . In addition to his major subjects, he also attended lectures in chemistry , zoology , geology , mineralogy and astronomy , as well as logic and metaphysics . In 1889 he continued his studies at the University of Heidelberg and received a doctoral degree in mathematics in 1892 (at the age of 22). His thesis is titled Zum Fundamentalsatz über die Existenz von Integralen partieller Differentialgleichungen (On

4515-402: The affine Lie algebras . For example to describe the su(2) subalgebras, the root system and the highest weight modules can be constructed by means of the ladder operators. In particular, the highest weight is annihilated by the raising operators; the rest of the positive root space is obtained by repeatedly applying the lowering operators (one set of ladder operators per subalgebra). From

4620-524: The anharmonic oscillator , with Hamiltonian H = 1 2 P 2 + 1 2 X 2 + ε X 3   . {\displaystyle H={1 \over 2}P^{2}+{1 \over 2}X^{2}+\varepsilon X^{3}~.} In this case, the X and P matrices are no longer simple off-diagonal matrices, since the corresponding classical orbits are slightly squashed and displaced, so that they have Fourier coefficients at every classical frequency. To determine

4725-419: The complex conjugates of the ones with positive frequencies, so that X ( t ) will always be real, X n = X − n ∗ . {\displaystyle X_{n}=X_{-n}^{*}.} A quantum mechanical particle, on the other hand, cannot emit radiation continuously; it can only emit photons. Assuming that the quantum particle started in orbit number n , emitted

4830-558: The quantum harmonic oscillator or the "number representation" of second quantization are just special cases of this fact. Ladder operators then become ubiquitous in quantum mechanics from the angular momentum operator , to coherent states and to discrete magnetic translation operators. Suppose that two operators X and N have the commutation relation [ N , X ] = c X {\displaystyle [N,X]=cX} for some scalar c . If | n ⟩ {\displaystyle {|n\rangle }}

4935-846: The "?" indicates a nascent quantum number which emerges from the discussion. Given the Pauli equations IV: 1 − A ⋅ A = − ( 2 E μ Z 2 e 4 ) ( L 2 + ℏ 2 ) {\displaystyle 1-A\cdot A=-\left({\frac {2E}{\mu Z^{2}e^{4}}}\right)(L^{2}+\hbar ^{2})} and III: ( A × A ) j = − ( 2 i ℏ E μ Z 2 e 4 ) L j , {\displaystyle \left(A\times A\right)_{j}=-\left({\frac {2{\boldsymbol {i}}\hbar E}{\mu Z^{2}e^{4}}}\right)L_{j},} and starting with

5040-587: The Bohr school, which was concerned with discrete energy states and quantum jumps. Bohr's followers did not appreciate physical models that pictured electrons as waves, or as anything at all. They preferred to focus on the quantities that were directly connected to experiments. In atomic physics, spectroscopy gave observational data on atomic transitions arising from the interactions of atoms with light quanta . The Bohr school required that only those quantities that were in principle measurable by spectroscopy should appear in

5145-508: The Fourier coefficients give the intensity of the emitted radiation , so in quantum mechanics the magnitude of the matrix elements of the position operator were the intensity of radiation in the bright-line spectrum. The quantities in Heisenberg's formulation were the classical position and momentum, but now they were no longer sharply defined. Each quantity was represented by a collection of Fourier coefficients with two indices, corresponding to

5250-412: The Fourier coefficients of a sharp classical trajectory . Nevertheless, as matrices, X ( t ) and P ( t ) satisfy the classical equations of motion; also see Ehrenfest's theorem, below. When it was introduced by Werner Heisenberg, Max Born and Pascual Jordan in 1925, matrix mechanics was not immediately accepted and was a source of controversy, at first. Schrödinger's later introduction of wave mechanics

5355-460: The Hamiltonian as H l {\displaystyle H_{l}} : H l = 1 2 μ [ p r 2 + 1 r 2 l ( l + 1 ) ℏ 2 ] + V ( r ) . {\displaystyle H_{l}={\frac {1}{2\mu }}\left[p_{r}^{2}+{\frac {1}{r^{2}}}l(l+1)\hbar ^{2}\right]+V(r).} The factorization method

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5460-809: The Hamiltonian for an inverse square spherically symmetric potential and can be used to determine ladder operators for this potential. We can define the lowering and raising operators (based on the classical Laplace–Runge–Lenz vector ) A → = ( 1 Z e 2 μ ) { L → × p → − i ℏ p → } + r → r , {\displaystyle {\vec {A}}=\left({\frac {1}{Ze^{2}\mu }}\right)\left\{{\vec {L}}\times {\vec {p}}-{\boldsymbol {i}}\hbar {\vec {p}}\right\}+{\frac {\vec {r}}{r}},} where L → {\displaystyle {\vec {L}}}

5565-1142: The Hamiltonian only has positive energy levels as can be seen from ⟨ ψ | 2 μ H l | ψ ⟩ = ⟨ ψ | C l ∗ C l | ψ ⟩ + ⟨ ψ | ( 2 l + 3 ) μ ω ℏ | ψ ⟩ = ⟨ C l ψ | C l ψ ⟩ + ( 2 l + 3 ) μ ω ℏ ⟨ ψ | ψ ⟩ ≥ 0. {\displaystyle {\begin{aligned}\langle \psi |2\mu H_{l}|\psi \rangle &=\langle \psi |C_{l}^{*}C_{l}|\psi \rangle +\langle \psi |(2l+3)\mu \omega \hbar |\psi \rangle \\&=\langle C_{l}\psi |C_{l}\psi \rangle +(2l+3)\mu \omega \hbar \langle \psi |\psi \rangle \\&\geq 0.\end{aligned}}} This means that for some value of l {\displaystyle l}

5670-414: The Hamiltonian, where l {\displaystyle l} is the angular momentum, and n {\displaystyle n} represents the energy, so L 2 | n l ⟩ = l ( l + 1 ) ℏ 2 | n l ⟩ {\displaystyle L^{2}|nl\rangle =l(l+1)\hbar ^{2}|nl\rangle } , and we may label

5775-572: The Nobel Prize in Physics for 1932 was delayed until November 1933. It was at that time that it was announced Heisenberg had won the Prize for 1932 "for the creation of quantum mechanics, the application of which has, inter alia , led to the discovery of the allotropic forms of hydrogen" and Erwin Schrödinger and Paul Adrien Maurice Dirac shared the 1933 Prize "for the discovery of new productive forms of atomic theory". It might well be asked why Born

5880-851: The above scalar product can be expanded as I ( 1 ) ⋅ J ( 1 ) = ∑ n = − 1 + 1 ( − 1 ) n I n ( 1 ) J − n ( 1 ) = I 0 ( 1 ) J 0 ( 1 ) − I − 1 ( 1 ) J + 1 ( 1 ) − I + 1 ( 1 ) J − 1 ( 1 ) . {\displaystyle \mathbf {I} ^{(1)}\cdot \mathbf {J} ^{(1)}=\sum _{n=-1}^{+1}(-1)^{n}I_{n}^{(1)}J_{-n}^{(1)}=I_{0}^{(1)}J_{0}^{(1)}-I_{-1}^{(1)}J_{+1}^{(1)}-I_{+1}^{(1)}J_{-1}^{(1)}.} The significance of this expansion

5985-816: The choice of units, are the Heisenberg matrices for the harmonic oscillator. Both matrices are hermitian , since they are constructed from the Fourier coefficients of real quantities. Finding X ( t ) and P ( t ) is direct, since they are quantum Fourier coefficients so they evolve simply with time, X m n ( t ) = X m n ( 0 ) e i ( E m − E n ) t , P m n ( t ) = P m n ( 0 ) e i ( E m − E n ) t   . {\displaystyle X_{mn}(t)=X_{mn}(0)e^{i(E_{m}-E_{n})t},\quad P_{mn}(t)=P_{mn}(0)e^{i(E_{m}-E_{n})t}~.} The matrix product of X and P

6090-435: The clockwise orbits, and they are nested circles in phase space. The classical orbit with energy E is X ( t ) = 2 E cos ⁡ ( t ) , P ( t ) = − 2 E sin ⁡ ( t )   . {\displaystyle X(t)={\sqrt {2E}}\cos(t),\qquad P(t)=-{\sqrt {2E}}\sin(t)~.} The old quantum condition dictates that

6195-441: The commutation relation [ N , X † ] = − c X † . {\displaystyle [N,X^{\dagger }]=-cX^{\dagger }.} In particular, if X is a lowering operator for N , then X is a raising operator for N and conversely. A particular application of the ladder operator concept is found in the quantum-mechanical treatment of angular momentum . For

6300-753: The condition that X is real becomes X n m = X m n ∗ . {\displaystyle X_{nm}=X_{mn}^{*}.} By definition, X nm only has the frequency ( E n − E m )/ h , so its time evolution is simple: X n m ( t ) = e 2 π i ( E n − E m ) t / h X n m ( 0 ) = e i ( E n − E m ) t / ℏ X n m ( 0 ) . {\displaystyle X_{nm}(t)=e^{2\pi i(E_{n}-E_{m})t/h}X_{nm}(0)=e^{i(E_{n}-E_{m})t/\hbar }X_{nm}(0).} This

6405-424: The correspondence with Fourier series allowed Heisenberg to deduce the rule by which the arrays should be multiplied, ( X P ) m n = ∑ k = 0 ∞ X m k P k n . {\displaystyle (XP)_{mn}=\sum _{k=0}^{\infty }X_{mk}P_{kn}.} Born pointed out that this is the law of matrix multiplication , so that

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6510-414: The corresponding eigenvector is the state of the system immediately after the measurement. The act of measurement in matrix mechanics 'collapses' the state of the system. If one measures two observables simultaneously, the state of the system collapses to a common eigenvector of the two observables. Since most matrices don't have any eigenvectors in common, most observables can never be measured precisely at

6615-424: The creation/annihilation operators of QFT requires the use of both annihilation and creation operators. An annihilation operator is used to remove a particle from the initial state and a creation operator is used to add a particle to the final state. The term "ladder operator" or "raising and lowering operators" is also sometimes used in mathematics, in the context of the theory of Lie algebras and in particular

6720-1908: The defining feature of ladder operators in quantum mechanics: the incrementing (or decrementing) of a quantum number, thus mapping one quantum state onto another. This is the reason that they are often known as raising and lowering operators. To obtain the values of α and β , first take the norm of each operator, recognizing that J + and J − are a Hermitian conjugate pair ( J ± = J ∓ † {\displaystyle J_{\pm }=J_{\mp }^{\dagger }} ): ⟨ j m | J + † J + | j m ⟩ = ⟨ j m | J − J + | j m ⟩ = ⟨ j ( m + 1 ) | α ∗ α | j ( m + 1 ) ⟩ = | α | 2 , ⟨ j m | J − † J − | j m ⟩ = ⟨ j m | J + J − | j m ⟩ = ⟨ j ( m − 1 ) | β ∗ β | j ( m − 1 ) ⟩ = | β | 2 . {\displaystyle {\begin{aligned}&\langle j\,m|J_{+}^{\dagger }J_{+}|j\,m\rangle =\langle j\,m|J_{-}J_{+}|j\,m\rangle =\langle j\,(m+1)|\alpha ^{*}\alpha |j\,(m+1)\rangle =|\alpha |^{2},\\&\langle j\,m|J_{-}^{\dagger }J_{-}|j\,m\rangle =\langle j\,m|J_{+}J_{-}|j\,m\rangle =\langle j\,(m-1)|\beta ^{*}\beta |j\,(m-1)\rangle =|\beta |^{2}.\end{aligned}}} The product of

6825-457: The discrete energy states and quantum jumps that Bohr discovered. De Broglie had reproduced the discrete energy states within Einstein's framework – the quantum condition is the standing wave condition, and this gave hope to those in the Einstein school that all the discrete aspects of quantum mechanics would be subsumed into a continuous wave mechanics. Matrix mechanics, on the other hand, came from

6930-606: The energy is an integer. The Fourier components of X ( t ) and P ( t ) are simple, and more so if they are combined into the quantities A ( t ) = X ( t ) + i P ( t ) = 2 E e − i t , A † ( t ) = X ( t ) − i P ( t ) = 2 E e i t . {\displaystyle A(t)=X(t)+iP(t)={\sqrt {2E}}\,e^{-it},\quad A^{\dagger }(t)=X(t)-iP(t)={\sqrt {2E}}\,e^{it}.} Both A and A have only

7035-1058: The equation A − A + | ℓ ∗ , ℓ ∗ ⟩ = 0 {\displaystyle A_{-}A_{+}|\ell ^{*},\ell ^{*}\rangle =0} and expanding, one obtains (assuming ℓ ∗ {\displaystyle \ell ^{*}} is the maximum value of the angular momentum quantum number consonant with all other conditions) ( 1 + 2 E μ Z 2 e 4 ( L 2 + ℏ 2 ) − i 2 i ℏ E μ Z 2 e 4 L z ) | ? , ℓ ∗ , ℓ ∗ ⟩ = 0 , {\displaystyle \left(1+{\frac {2E}{\mu Z^{2}e^{4}}}(L^{2}+\hbar ^{2})-i{\frac {2i\hbar E}{\mu Z^{2}e^{4}}}L_{z}\right)|?,\ell ^{*},\ell ^{*}\rangle =0,} which leads to

7140-1614: The factorization method. A suitable factorization is given by C l = p r + i ℏ ( l + 1 ) r − i μ ω r {\displaystyle C_{l}=p_{r}+{\frac {i\hbar (l+1)}{r}}-i\mu \omega r} with F l = − ( 2 l + 3 ) μ ω ℏ {\displaystyle F_{l}=-(2l+3)\mu \omega \hbar } and G l = − ( 2 l + 1 ) μ ω ℏ . {\displaystyle G_{l}=-(2l+1)\mu \omega \hbar .} Then E l + 1 n ′ = E l n + F l − G l 2 μ = E l n − ω ℏ , {\displaystyle E_{l+1}^{n^{'}}=E_{l}^{n}+{\frac {F_{l}-G_{l}}{2\mu }}=E_{l}^{n}-\omega \hbar ,} and continuing this, E l + 2 n ′ = E l n − 2 ω ℏ E l + 3 n ′ = E l n − 3 ω ℏ ⋮ {\displaystyle {\begin{aligned}E_{l+2}^{n^{'}}&=E_{l}^{n}-2\omega \hbar \\E_{l+3}^{n^{'}}&=E_{l}^{n}-3\omega \hbar \\&\;\;\vdots \end{aligned}}} Now

7245-400: The final mathematical formulation of matrix mechanics and Heisenberg went on to stress how great their contributions were to quantum mechanics, which were not "adequately acknowledged in the public eye". Once Heisenberg introduced the matrices for X and P , he could find their matrix elements in special cases by guesswork, guided by the correspondence principle. Since the matrix elements are

7350-403: The first attempt at a unified theory of matter in the 20th century. His motivation was to explain the 'invisible' electron and relate gravitation to matter; his theory had three core assumptions: 1) electrical and magnetic fields exist inside of electrons, 2) special relativity, and 3) new states of ether would be sufficient to explain all phenomenon of the material world. A crater on Mars

7455-414: The formula above, T is the classical period of either orbit n or orbit m , since the difference between them is higher order in h . But for n and m small, or if n − m is large, the frequencies are not integer multiples of any single frequency. Since the frequencies that the particle emits are the same as the frequencies in the Fourier description of its motion, this suggests that something in

7560-530: The fundamental theorem on the existence of integrals of partial differential equations) and his supervisor was Leo Königsberger . In 1897 he got his habilitation at the University of Göttingen in theoretical physics and in 1902 became extraordinary professor for theoretical physics at the University of Greifswald . In 1917 he became full professor for experimental physics at Martin Luther University of Halle-Wittenberg . In 1924 he became professor at

7665-440: The initial and final states. When Born read the paper, he recognized the formulation as one which could be transcribed and extended to the systematic language of matrices , which he had learned from his study under Jakob Rosanes at Breslau University . Born, with the help of his assistant and former student Pascual Jordan, began immediately to make the transcription and extension, and they submitted their results for publication;

7770-401: The integral of P dX over an orbit, which is the area of the circle in phase space, must be an integer multiple of the Planck constant . The area of the circle of radius √ 2 E is 2 πE . So E = n h 2 π = n ℏ , {\displaystyle E={\frac {nh}{2\pi }}=n\hbar \,,} or, in natural units where ħ = 1 ,

7875-1250: The ladder operators can be expressed in terms of the commuting pair J and J z : J − J + = ( J x − i J y ) ( J x + i J y ) = J x 2 + J y 2 + i [ J x , J y ] = J 2 − J z 2 − ℏ J z , J + J − = ( J x + i J y ) ( J x − i J y ) = J x 2 + J y 2 − i [ J x , J y ] = J 2 − J z 2 + ℏ J z . {\displaystyle {\begin{aligned}J_{-}J_{+}&=(J_{x}-iJ_{y})(J_{x}+iJ_{y})=J_{x}^{2}+J_{y}^{2}+i[J_{x},J_{y}]=J^{2}-J_{z}^{2}-\hbar J_{z},\\J_{+}J_{-}&=(J_{x}+iJ_{y})(J_{x}-iJ_{y})=J_{x}^{2}+J_{y}^{2}-i[J_{x},J_{y}]=J^{2}-J_{z}^{2}+\hbar J_{z}.\end{aligned}}} Thus, one may express

7980-400: The language and framework usually employed today, in full display of the noncommutative structure of the entire construction. Before matrix mechanics, the old quantum theory described the motion of a particle by a classical orbit, with well defined position and momentum X ( t ), P ( t ), with the restriction that the time integral over one period T of the momentum times the velocity must be

8085-549: The lowering operators C ∗ {\displaystyle C^{*}} : C n − 2 ∗ | n − 1 , n − 1 ⟩ , C n − 4 ∗ C n − 3 ∗ | n − 2 , n − 2 ⟩ , … {\displaystyle C_{n-2}^{*}|n-1,\,n-1\rangle ,C_{n-4}^{*}C_{n-3}^{*}|n-2,\,n-2\rangle ,\dots } giving

8190-472: The matrix elements, Heisenberg required that the classical equations of motion be obeyed as matrix equations, d X d t = P d P d t = − X − 3 ε X 2   . {\displaystyle {dX \over dt}=P\quad {dP \over dt}=-X-3\varepsilon X^{2}~.} Ladder operator In linear algebra (and its application to quantum mechanics ),

8295-401: The old quantum mechanical formalism did not describe time dependent processes, such as the emission or absorption of radiation. When a classical particle is weakly coupled to a radiation field, so that the radiative damping can be neglected, it will emit radiation in a pattern that repeats itself every orbital period . The frequencies that make up the outgoing wave are then integer multiples of

8400-521: The orbital frequency, and this is a reflection of the fact that X ( t ) is periodic, so that its Fourier representation has frequencies 2π n / T only. X ( t ) = ∑ n = − ∞ ∞ e 2 π i n t / T X n . {\displaystyle X(t)=\sum _{n=-\infty }^{\infty }e^{2\pi int/T}X_{n}.} The coefficients X n are complex numbers . The ones with negative frequencies must be

8505-831: The paper was received for publication just 60 days after Heisenberg's paper. A follow-on paper was submitted for publication before the end of the year by all three authors. (A brief review of Born's role in the development of the matrix mechanics formulation of quantum mechanics along with a discussion of the key formula involving the non-commutativity of the probability amplitudes can be found in an article by Jeremy Bernstein . A detailed historical and technical account can be found in Mehra and Rechenberg's book The Historical Development of Quantum Theory. Volume 3. The Formulation of Matrix Mechanics and Its Modifications 1925–1926. ) The three fundamental papers: Up until this time, matrices were seldom used by physicists; they were considered to belong to

8610-547: The paper, Heisenberg formulated quantum theory without sharp electron orbits. Hendrik Kramers had earlier calculated the relative intensities of spectral lines in the Sommerfeld model by interpreting the Fourier coefficients of the orbits as intensities. But his answer, like all other calculations in the old quantum theory , was only correct for large orbits . Heisenberg, after a collaboration with Kramers, began to understand that

8715-455: The particle size and explain the change in colour that occurs as the size of the colloid nanoparticles is increased from 20 to 1600 nm. He wrote further important contributions to electromagnetism and also to relativity theory . In addition he was employed on measurements units and finally developed his Mie system of units in 1910 with the basic units Volt , Ampere , Coulomb and Second (VACS-system). In 1912 and 1913 Mie published

8820-432: The position, the momentum, the energy, all the observable quantities in the theory, are interpreted as matrices. Under this multiplication rule, the product depends on the order: XP is different from PX . The X matrix is a complete description of the motion of a quantum mechanical particle. Because the frequencies in the quantum motion are not multiples of a common frequency, the matrix elements cannot be interpreted as

8925-546: The quantum mechanical analogs of Fourier coefficients of the classical orbits, the simplest case is the harmonic oscillator , where the classical position and momentum, X ( t ) and P ( t ), are sinusoidal. In units where the mass and frequency of the oscillator are equal to one (see nondimensionalization ), the energy of the oscillator is H = 1 2 ( P 2 + X 2 ) . {\displaystyle H={1 \over 2}\left(P^{2}+X^{2}\right).} The level sets of H are

9030-418: The radial momentum p r = x r p x + y r p y + z r p z , {\displaystyle p_{r}={\frac {x}{r}}p_{x}+{\frac {y}{r}}p_{y}+{\frac {z}{r}}p_{z},} which is real and self-conjugate. Suppose | n l ⟩ {\displaystyle |nl\rangle } is an eigenvector of

9135-551: The realm of pure mathematics. Gustav Mie had used them in a paper on electrodynamics in 1912 and Born had used them in his work on the lattices theory of crystals in 1921. While matrices were used in these cases, the algebra of matrices with their multiplication did not enter the picture as they did in the matrix formulation of quantum mechanics. Born, however, had learned matrix algebra from Rosanes, as already noted, but Born had also learned Hilbert's theory of integral equations and quadratic forms for an infinite number of variables as

9240-405: The requirements of the above definition of a ladder operator: the incrementing or decrementing of the eigenvalue of another operator (in this case the particle number operator ). Confusion arises because the term ladder operator is typically used to describe an operator that acts to increment or decrement a quantum number describing the state of a system. To change the state of a particle with

9345-915: The same time. This is the uncertainty principle . If two matrices share their eigenvectors, they can be simultaneously diagonalized. In the basis where they are both diagonal, it is clear that their product does not depend on their order because multiplication of diagonal matrices is just multiplication of numbers. The uncertainty principle, by contrast, is an expression of the fact that often two matrices A and B do not always commute, i.e., that AB − BA does not necessarily equal 0. The fundamental commutation relation of matrix mechanics, ∑ k ( X n k P k m − P n k X k m ) = i ℏ δ n m {\displaystyle \sum _{k}(X_{nk}P_{km}-P_{nk}X_{km})=i\hbar \,\delta _{nm}} implies then that there are no states that simultaneously have

9450-436: The sequence | n , n ⟩ , | n , n − 2 ⟩ , | n , n − 4 ⟩ , … {\displaystyle |n,n\rangle ,|n,\,n-2\rangle ,|n,\,n-4\rangle ,\dots } with the same energy but with l {\displaystyle l} decreasing by 2. In addition to the angular momentum degeneracy, this gives

9555-599: The series must terminate with C l max | n l max ⟩ = 0 , {\displaystyle C_{l_{\text{max}}}|nl_{\text{max}}\rangle =0,} and then E l max n = − F l max 2 μ = ( l max + 3 2 ) ω ℏ . {\displaystyle E_{l_{\text{max}}}^{n}=-{\frac {F_{l_{\text{max}}}}{2\mu }}=\left(l_{\text{max}}+{\frac {3}{2}}\right)\omega \hbar .} This

9660-492: The spectral issue and eventually realised that adopting non-commuting observables might solve the problem. He later wrote: It was about three o' clock at night when the final result of the calculation lay before me. At first I was deeply shaken. I was so excited that I could not think of sleep. So I left the house and awaited the sunrise on the top of a rock. After Heisenberg returned to Göttingen, he showed Wolfgang Pauli his calculations, commenting at one point: Everything

9765-562: The states | n l ⟩ {\displaystyle |nl\rangle } and C l | n l ⟩ {\displaystyle C_{l}|nl\rangle } have the same energy. For the hydrogenic atom, setting V ( r ) = − B ℏ μ r {\displaystyle V(r)=-{\frac {B\hbar }{\mu r}}} with B = Z μ e 2 ℏ , {\displaystyle B={\frac {Z\mu e^{2}}{\hbar }},}

9870-414: The theory. These quantities include the energy levels and their intensities but they do not include the exact location of a particle in its Bohr orbit. It is very hard to imagine an experiment that could determine whether an electron in the ground state of a hydrogen atom is to the right or to the left of the nucleus. It was a deep conviction that such questions did not have an answer. The matrix formulation

9975-457: The time-dependent description of the particle is oscillating with frequency ( E n − E m )/ h . Heisenberg called this quantity X nm , and demanded that it should reduce to the classical Fourier coefficients in the classical limit. For large values of n , m but with n − m relatively small, X nm is the ( n − m ) th Fourier coefficient of the classical motion at orbit n . Since X nm has opposite frequency to X mn ,

10080-399: The transition probabilities were not quite classical quantities, because the only frequencies that appear in the Fourier series should be the ones that are observed in quantum jumps, not the fictional ones that come from Fourier-analyzing sharp classical orbits. He replaced the classical Fourier series with a matrix of coefficients, a fuzzed-out quantum analog of the Fourier series. Classically,

10185-556: The values x , y and z . From this, the commutation relations among the ladder operators and J z are obtained: [ J z , J ± ] = ± ℏ J ± , [ J + , J − ] = 2 ℏ J z {\displaystyle {\begin{aligned}{}[J_{z},J_{\pm }]&=\pm \hbar J_{\pm },\\{}[J_{+},J_{-}]&=2\hbar J_{z}\end{aligned}}} (technically, this

10290-2221: The values of | α | and | β | in terms of the eigenvalues of J and J z : | α | 2 = ℏ 2 j ( j + 1 ) − ℏ 2 m 2 − ℏ 2 m = ℏ 2 ( j − m ) ( j + m + 1 ) , | β | 2 = ℏ 2 j ( j + 1 ) − ℏ 2 m 2 + ℏ 2 m = ℏ 2 ( j + m ) ( j − m + 1 ) . {\displaystyle {\begin{aligned}|\alpha |^{2}&=\hbar ^{2}j(j+1)-\hbar ^{2}m^{2}-\hbar ^{2}m=\hbar ^{2}(j-m)(j+m+1),\\|\beta |^{2}&=\hbar ^{2}j(j+1)-\hbar ^{2}m^{2}+\hbar ^{2}m=\hbar ^{2}(j+m)(j-m+1).\end{aligned}}} The phases of α and β are not physically significant, thus they can be chosen to be positive and real ( Condon–Shortley phase convention ). We then have J + | j , m ⟩ = ℏ ( j − m ) ( j + m + 1 ) | j , m + 1 ⟩ = ℏ j ( j + 1 ) − m ( m + 1 ) | j , m + 1 ⟩ , J − | j , m ⟩ = ℏ ( j + m ) ( j − m + 1 ) | j , m − 1 ⟩ = ℏ j ( j + 1 ) − m ( m − 1 ) | j , m − 1 ⟩ . {\displaystyle {\begin{aligned}J_{+}|j,m\rangle &=\hbar {\sqrt {(j-m)(j+m+1)}}|j,m+1\rangle =\hbar {\sqrt {j(j+1)-m(m+1)}}|j,m+1\rangle ,\\J_{-}|j,m\rangle &=\hbar {\sqrt {(j+m)(j-m+1)}}|j,m-1\rangle =\hbar {\sqrt {j(j+1)-m(m-1)}}|j,m-1\rangle .\end{aligned}}} Confirming that m

10395-434: Was apparent from a citation by Born of Hilbert's work Grundzüge einer allgemeinen Theorie der Linearen Integralgleichungen published in 1912. Jordan, too, was well equipped for the task. For a number of years, he had been an assistant to Richard Courant at Göttingen in the preparation of Courant and David Hilbert 's book Methoden der mathematischen Physik I , which was published in 1924. This book, fortuitously, contained

10500-417: Was built on the premise that all physical observables are represented by matrices, whose elements are indexed by two different energy levels. The set of eigenvalues of the matrix were eventually understood to be the set of all possible values that the observable can have. Since Heisenberg's matrices are Hermitian , the eigenvalues are real. If an observable is measured and the result is a certain eigenvalue,

10605-2018: Was developed by Infeld and Hull for differential equations. Newmarch and Golding applied it to spherically symmetric potentials using operator notation. Suppose we can find a factorization of the Hamiltonian by operators C l {\displaystyle C_{l}} as and C l C l ∗ = 2 μ H l + 1 + G l {\displaystyle C_{l}C_{l}^{*}=2\mu H_{l+1}+G_{l}} for scalars F l {\displaystyle F_{l}} and G l {\displaystyle G_{l}} . The vector C l C l ∗ C l | n l ⟩ {\displaystyle C_{l}C_{l}^{*}C_{l}|nl\rangle } may be evaluated in two different ways as C l C l ∗ C l | n l ⟩ = ( 2 μ E l n + F l ) C l | n l ⟩ = ( 2 μ H l + 1 + G l ) C l | n l ⟩ , {\displaystyle {\begin{aligned}C_{l}C_{l}^{*}C_{l}|nl\rangle &=(2\mu E_{l}^{n}+F_{l})C_{l}|nl\rangle \\&=(2\mu H_{l+1}+G_{l})C_{l}|nl\rangle ,\end{aligned}}} which can be re-arranged as H l + 1 ( C l | n l ⟩ ) = [ E l n + ( F l − G l ) / ( 2 μ ) ] ( C l | n l ⟩ ) , {\displaystyle H_{l+1}(C_{l}|nl\rangle )=[E_{l}^{n}+(F_{l}-G_{l})/(2\mu )](C_{l}|nl\rangle ),} showing that C l | n l ⟩ {\displaystyle C_{l}|nl\rangle }

10710-427: Was greatly favored. Part of the reason was that Heisenberg's formulation was in an odd mathematical language, for the time, while Schrödinger's formulation was based on familiar wave equations. But there was also a deeper sociological reason. Quantum mechanics had been developing by two paths, one led by Einstein, who emphasized the wave–particle duality he proposed for photons, and the other led by Bohr, that emphasized

10815-459: Was not awarded the Prize in 1932, along with Heisenberg, and Bernstein proffers speculations on this matter. One of them relates to Jordan joining the Nazi Party on May 1, 1933, and becoming a stormtrooper . Jordan's Party affiliations and Jordan's links to Born may well have affected Born's chance at the Prize at that time. Bernstein further notes that when Born finally won the Prize in 1954, Jordan

10920-457: Was still alive, while the Prize was awarded for the statistical interpretation of quantum mechanics, attributable to Born alone. Heisenberg's reactions to Born for Heisenberg receiving the Prize for 1932 and for Born receiving the Prize in 1954 are also instructive in evaluating whether Born should have shared the Prize with Heisenberg. On November 25, 1933, Born received a letter from Heisenberg in which he said he had been delayed in writing due to

11025-500: Was working in Göttingen on the problem of calculating the spectral lines of hydrogen . By May 1925 he began trying to describe atomic systems by observables only. On June 7, after weeks of failing to alleviate his hay fever with aspirin and cocaine, Heisenberg left for the pollen-free North Sea island of Helgoland . While there, in between climbing and memorizing poems from Goethe 's West-östlicher Diwan , he continued to ponder

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