In physics, the
Contents 1 Mathematical details
2 Equivalence of Heisenberg's equation to the Schrödinger equation
3
Mathematical details[edit]
In the
d d t A ( t ) = i ℏ [ H , A ( t ) ] + ( ∂ A ∂ t ) H , displaystyle frac d dt A(t)= frac i hbar [H,A(t)]+left( frac partial A partial t right)_ H , where H is the Hamiltonian and [•,•] denotes the commutator of two
operators (in this case H and A). Taking expectation values
automatically yields the Ehrenfest theorem, featured in the
correspondence principle.
By the Stone–von Neumann theorem, the
⟨ A ⟩ t = ⟨ ψ ( t )
A
ψ ( t ) ⟩ . displaystyle langle Arangle _ t =langle psi (t)Apsi (t)rangle . In the Schrödinger picture, the state ψ(t)〉at time t is related to the state ψ(0)〉at time 0 by a unitary time-evolution operator, U(t),
ψ ( t ) ⟩ = U ( t )
ψ ( 0 ) ⟩ . displaystyle psi (t)rangle =U(t)psi (0)rangle . If the Hamiltonian does not vary with time, then the time-evolution operator can be written as U ( t ) = e − i H t / ℏ , displaystyle U(t)=e^ -iHt/hbar , where H is the Hamiltonian and ħ is the reduced Planck constant. Therefore, ⟨ A ⟩ t = ⟨ ψ ( 0 )
e + i H t / ℏ A e − i H t / ℏ
ψ ( 0 ) ⟩ . displaystyle langle Arangle _ t =langle psi (0)e^ +iHt/hbar Ae^ -iHt/hbar psi (0)rangle . Peg all state vectors to a rigid basis of ψ(0)〉, and then define A ( t ) := e + i H t / ℏ A e − i H t / ℏ . displaystyle A(t):=e^ +iHt/hbar Ae^ -iHt/hbar . It now follows that d d t A ( t ) = i ℏ H e i H t / ℏ A e − i H t / ℏ + e + i H t / ℏ ( ∂ A ∂ t ) e − i H t / ℏ + i ℏ e + i H t / ℏ A ⋅ ( − H ) e − i H t / ℏ = i ℏ e i H t / ℏ ( H A − A H ) e − i H t / ℏ + e + i H t / ℏ ( ∂ A ∂ t ) e − i H t / ℏ = i ℏ ( H A ( t ) − A ( t ) H ) + e + i H t / ℏ ( ∂ A ∂ t ) e − i H t / ℏ . displaystyle begin aligned operatorname d over operatorname d !t A(t)&= i over hbar He^ iHt/hbar Ae^ -iHt/hbar +e^ +iHt/hbar left( frac partial A partial t right)e^ -iHt/hbar + i over hbar e^ +iHt/hbar Acdot (-H)e^ -iHt/hbar \&= i over hbar e^ iHt/hbar left(HA-AHright)e^ -iHt/hbar +e^ +iHt/hbar left( frac partial A partial t right)e^ -iHt/hbar \&= i over hbar left(HA(t)-A(t)Hright)+e^ +iHt/hbar left( frac partial A partial t right)e^ -iHt/hbar .end aligned Differentiation was according to the product rule, while ∂A/∂t is the time derivative of the initial A, not the A(t) operator defined. The last equation holds since exp(−i H t/ħ) commutes with H. Thus d d t A ( t ) = i ℏ [ H , A ( t ) ] + e + i H t / ℏ ( ∂ A ∂ t ) e − i H t / ℏ , displaystyle operatorname d over operatorname d !t A(t)= i over hbar [H,A(t)]+e^ +iHt/hbar left( frac partial A partial t right)e^ -iHt/hbar , and hence emerges the above
e B A e − B = A + [ B , A ] + 1 2 ! [ B , [ B , A ] ] + 1 3 ! [ B , [ B , [ B , A ] ] ] + ⋯ . displaystyle e^ B Ae^ -B =A+[B,A]+ frac 1 2! [B,[B,A]]+ frac 1 3! [B,[B,[B,A]]]+cdots . which implies A ( t ) = A + i t ℏ [ H , A ] + 1 2 ! ( i t ℏ ) 2 [ H , [ H , A ] ] + 1 3 ! ( i t ℏ ) 3 [ H , [ H , [ H , A ] ] ] + … displaystyle A(t)=A+ frac it hbar [H,A]+ frac 1 2! left( frac it hbar right)^ 2 [H,[H,A]]+ frac 1 3! left( frac it hbar right)^ 3 [H,[H,[H,A]]]+dots This relation also holds for classical mechanics, the classical limit of the above, given the correspondence between Poisson brackets and commutators, [ A , H ] ⟷ i ℏ A , H displaystyle [A,H]quad longleftrightarrow quad ihbar A,H In classical mechanics, for an A with no explicit time dependence, A , H = d A d t , displaystyle A,H = frac operatorname d !A operatorname d !t ~, so again the expression for A(t) is the Taylor expansion around t = 0.
In effect, the arbitrary rigid
H = p 2 2 m + m ω 2 x 2 2 displaystyle H= frac p^ 2 2m + frac momega ^ 2 x^ 2 2 , the evolution of the position and momentum operators is given by: d d t x ( t ) = i ℏ [ H , x ( t ) ] = p m displaystyle d over dt x(t)= i over hbar [H,x(t)]= frac p m , d d t p ( t ) = i ℏ [ H , p ( t ) ] = − m ω 2 x displaystyle d over dt p(t)= i over hbar [H,p(t)]=-momega ^ 2 x . Differentiating both equations once more and solving for them with proper initial conditions, p ˙ ( 0 ) = − m ω 2 x 0 , displaystyle dot p (0)=-momega ^ 2 x_ 0 , x ˙ ( 0 ) = p 0 m , displaystyle dot x (0)= frac p_ 0 m , leads to x ( t ) = x 0 cos ( ω t ) + p 0 ω m sin ( ω t ) displaystyle x(t)=x_ 0 cos(omega t)+ frac p_ 0 omega m sin(omega t) , p ( t ) = p 0 cos ( ω t ) − m ω x 0 sin ( ω t ) displaystyle p(t)=p_ 0 cos(omega t)-momega !x_ 0 sin(omega t) . Direct computation yields the more general commutator relations, [ x ( t 1 ) , x ( t 2 ) ] = i ℏ m ω sin ( ω t 2 − ω t 1 ) displaystyle [x(t_ 1 ),x(t_ 2 )]= frac ihbar momega sin(omega t_ 2 -omega t_ 1 ) , [ p ( t 1 ) , p ( t 2 ) ] = i ℏ m ω sin ( ω t 2 − ω t 1 ) displaystyle [p(t_ 1 ),p(t_ 2 )]=ihbar momega sin(omega t_ 2 -omega t_ 1 ) , [ x ( t 1 ) , p ( t 2 ) ] = i ℏ cos ( ω t 2 − ω t 1 ) displaystyle [x(t_ 1 ),p(t_ 2 )]=ihbar cos(omega t_ 2 -omega t_ 1 ) . For t 1 = t 2 displaystyle t_ 1 =t_ 2 , one simply recovers the standard canonical commutation relations valid in all pictures. Summary comparison of evolution in all pictures[edit] Evolution Picture of: Heisenberg Interaction Schrödinger Ket state constant
ψ I ( t ) ⟩ = e i H 0 , S t / ℏ
ψ S ( t ) ⟩ displaystyle psi _ I (t)rangle =e^ iH_ 0,S ~t/hbar psi _ S (t)rangle
ψ S ( t ) ⟩ = e − i H S t / ℏ
ψ S ( 0 ) ⟩ displaystyle psi _ S (t)rangle =e^ -iH_ S ~t/hbar psi _ S (0)rangle Observable A H ( t ) = e i H S t / ℏ A S e − i H S t / ℏ displaystyle A_ H (t)=e^ iH_ S ~t/hbar A_ S e^ -iH_ S ~t/hbar A I ( t ) = e i H 0 , S t / ℏ A S e − i H 0 , S t / ℏ displaystyle A_ I (t)=e^ iH_ 0,S ~t/hbar A_ S e^ -iH_ 0,S ~t/hbar constant Density matrix constant ρ I ( t ) = e i H 0 , S t / ℏ ρ S ( t ) e − i H 0 , S t / ℏ displaystyle rho _ I (t)=e^ iH_ 0,S ~t/hbar rho _ S (t)e^ -iH_ 0,S ~t/hbar ρ S ( t ) = e − i H S t / ℏ ρ S ( 0 ) e i H S t / ℏ displaystyle rho _ S (t)=e^ -iH_ S ~t/hbar rho _ S (0)e^ iH_ S ~t/hbar See also[edit] Bra–ket notation Interaction picture Schrödinger picture Heisenberg–Langevin equations Phase space formulation References[edit] ^ "
Cohen-Tannoudji, Claude; Bernard Diu; Frank Laloe (1977). Quantum
Mechanics (Volume One). Paris: Wiley. pp. 312–314.
ISBN 0-471-16433-X.
Albert Messiah, 1966.
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