p. 33
SCHRÖDINGER AND HEISENBERG REPRESENTATIONS
The mathematical formulation of the dynamics of a quantum system is not unique. So far we have described the dynamics by propagating the wavefunction, which encodes probability densities. This is known as the Schrödinger representation of quantum mechanics. Ultimately, since we can’t measure a wavefunction, we are interested in observables (probability amplitudes associated with Hermetian operators). Looking at a time-evolving expectation value suggests an alternate interpretation of the quantum observable:
Atˆ () = ψ ()t Aˆ ψ ()t = ψ ()0 U † AˆUψ ()0 = ( ψ ()0 UA† ) (U ψ ()0 ) (4.1) = ψ ()0 (UA † U) ψ ()0
The last two expressions here suggest alternate transformation that can describe the dynamics. These have different physical interpretations:
1) Transform the eigenvectors: ψ (t) →U ψ . Leave operators unchanged.
2) Transform the operators: Atˆ ( ) →U† AˆU. Leave eigenvectors unchanged.
(1) Schrödinger Picture: Everything we have done so far. Operators are stationary.
Eigenvectors evolve under Ut( ,t0 ) .
(2) Heisenberg Picture: Use unitary property of U to transform operators so they evolve in time. The wavefunction is stationary. This is a physically appealing picture, because particles move – there is a time-dependence to position and momentum.
Let’s look at time-evolution in these two pictures:
Schrödinger Picture
We have talked about the time-development of ψ , which is governed by
∂ i ψ = H ψ (4.2) ∂t p. 34
in differential form, or alternatively ψ (t) = U( t, t0 ) ψ (t0 ) in an integral form. In the ∂A Schrödinger picture, for operators typically = 0 . What about observables? For expectation ∂t values of operators Aˆ()t = ψ ()t Aˆ ψ ()t :
∂ ⎡ ∂ψ ∂ψ ∂A ˆ ⎤ i Aˆ()t= i ⎢ ψ Aˆ + Aˆ ψ + ψ ψ ⎥ ∂t ⎢ ∂t ∂t ∂t ⎥ ⎣ ⎦ ˆ ˆ = ψ AHψ − ψ H Aψ (4.3) ⎡ ˆ ⎤ = ψ ⎣AH, ⎦ ψ ⎡ ˆ ⎤ = ⎣AH, ⎦
Alternatively, written for the density matrix:
∂ ∂ i Aˆ()t= i T r( Aρ ) ∂t ∂t ⎛ ∂ ⎞ = iT r⎜ A ρ ⎟ ⎝ ∂t ⎠ (4.4) = Tr ( A[ H , ρ]) = Tr ( [ A, H ]ρ )
If A is independent of time (as we expect in the Schrödinger picture) and if it commutes with H , it is referred to as a constant of motion.
Heisenberg Picture
From eq. (4.1) we can distinguish the Schrödinger picture from Heisenberg operators:
ˆ ˆ † At()= ψ ()t A ψ ()t = ψ ()t0 U AUψ ()t0 = ψ A()t ψ (4.5) S S H where the operator is defined as
At( ) = U † (t, t) A U(t, t) H 0 S 0 (4.6) AtH ()0 = AS p. 35
Also, since the wavefunction should be time-independent ∂ ψ H ∂=t 0 , we can relate the Schrödinger and Heisenberg wavefunctions as
ψ S (t) = U(t,t0 ) ψ H (4.7)
† So, ψ H = Ut( ,t0 ) ψ S (t) = ψ S (t0 ) (4.8)
In either picture the eigenvalues are preserved:
A ϕ = a ϕ i S i i S UA† UU† ϕ = aU † ϕ (4.9) i S i i S A ϕ = a ϕ H i H i i H
The time-evolution of the operators in the Heisenberg picture is:
† ∂AH ∂ † ∂U † ∂U † ∂A S = (UAS U) = AS U+ UAS +U U ∂t ∂t ∂t ∂t ∂t
i † i † ⎛ ∂A ⎞ = UHAS U− UAS HU+ ⎜ ⎟ ∂t ⎝ ⎠ H (4.10) i i = HA− AH H H H H −i = [ AH, ] H
∂ The result: i A = [ A, H] (4.11) ∂t H H
† is known as the Heisenberg equation of motion. Here I have written H H = UHU. Generally
−iHt / speaking, for a time-independent Hamiltonian Ue= , U and H commute, and H H = H . For a time-dependent Hamiltonian, U and H need not commute.
Particle in a potential
Often we want to describe the equations of motion for particles with an arbitrary potential:
p 2 H = +V()x (4.12) 2m p. 36
For which the Heisenberg equation gives:
∂V p =− . (4.13) ∂x
p x = (4.14) m
n n−1 Here, I’ve made use of ⎣⎡xˆˆ, p⎦⎤ = i nxˆ (4.15)
n n−1 ⎣⎡xˆˆ, p ⎦⎤ = i npˆ (4.16)
These equations indicate that the position and momentum operators follow equations of motion identical to the classical variables in Hamilton’s equations. These do not involve factors of . Note here that if we integrate eq. (4.14) over a time period t we find:
pt xt() = + x()0 (4.17) m implying that the expectation value for the position of the particle follows the classical motion. These equations also hold for the expectation values for the position and momentum operators (Ehrenfest’s Theorem) and indicate the nature of the classical correspondence. In correspondence to Newton’s equation, we see
∂2 x m =− ∇V (4.18) ∂t 2 p. 37
The interaction picture is a hybrid representation that is useful in solving problems with time- dependent Hamiltonians in which we can partition the Hamiltonian as
Ht( ) = H0 +V(t) (4.19)
H0 is a Hamiltonian for the degrees of freedom we are interested in, which we treat exactly, and can be (although for us generally won’t be) a function of time. Vt( ) is a time-dependent potential which can be complicated. In the interaction picture we will treat each part of the Hamiltonian in a different representation. We will use the eigenstates of H0 as a basis set to describe the dynamics induced by Vt(), assuming that Vt( ) is small enough that eigenstates of H0 are a useful basis to describe H. If H0 is not a function of time, then there is simple time-dependence to this part of the Hamiltonian, that we may be able to account for easily.
Setting V to zero, we can see that the time evolution of the exact part of the Hamiltonian
H0 is described by
∂ −i Ut( , t) = H()t U (t , t) (4.20) ∂t 0 0 0 0 0
⎡ i t ⎤ where, most generally, Ut( , t) = exp dτ Ht() (4.21) 0 0 + ⎢ ∫t 0 ⎥ ⎣ 0 ⎦
−iH 0 (t−t0 ) but for a time-independent H0 Ut0 ( ,t0 ) = e (4.22)
We define a wavefunction in the interaction picture ψ I through:
ψ S (t) ≡ U0 (t,t 0 ) ψ I (t) (4.23)
† or ψ I = U0 ψ S (4.24)
Effectively the interaction representation defines wavefunctions in such a way that the phase accumulated under e−iH0t is removed. For small V, these are typically high frequency oscillations relative to the slower amplitude changes in coherences induced by V. p. 38
We are after an equation of motion that describes the time-evolution of the interaction picture wave-functions. We begin by substituting eq. (4.23) into the TDSE:
∂ i ψ = H ψ (4.25) ∂t S S
∂ −i Ut( ,t ) ψ = H()tU ( t,t ) ψ ∂t 0 0 I 0 0 I
∂U0 ∂ ψ I −i ψ I +U0= (H 0+ V()t) U0(t ,t 0`) ψ I (4.26) ∂t ∂t
−i ∂ ψ −i HUψ +U I = H +Vt()U ψ 0 0 I 0 ∂t ( 0 ) 0 I
∂ ψ ∴ i I = V ψ (4.27) ∂t I I
† where VtI ()= U0 ( t,t0 ) V(t) U0 ( t,t 0 ) (4.28)
ψ I satisfies the Schrödinger equation with a new Hamiltonian: the interaction picture
Hamiltonian, VtI (), which is the U0 unitary transformation of Vt( ). Note: Matrix elements in
−iωlkt VI = k VI l = e Vkl where k and l are eigenstates of H0.
We can now define a time-evolution operator in the interaction picture:
ψ I (t) = UI (t,t0 ) ψ I (t0 ) (4.29)
⎡−i t ⎤ where Ut( ,t) = exp dτ V()τ . (4.30) I 0 + ⎢ ∫t I ⎥ ⎣ 0 ⎦
Now we see that
ψ S ()t = U0 (t,t0 ) ψ I (t)
= Ut0 ( , t0 )UI (t, t 0 ) ψ I (t 0 ) (4.31)
= Ut0 ( , t0 )UI (t, t 0 ) ψ S (t0 )
∴ Ut( ,t 0 ) = U0 ( t,t0 ) UI ( t,t 0 ) (4.32) p. 39
Using the time ordered exponential in eq. (4.30), U can be written as
Ut(, t 0 )= U 0 (t,t 0 )+
∞ n ⎛ −i ⎞ t τ n τ 2 dτ dτ … dτ U(t,τ )V (τ )U (τ , τ ) … (4.33) ∑⎜ ⎟ ∫t n ∫t n−1 ∫t 1 0 n n 0 n n−1 n=1 ⎝ ⎠ 0 0 0
U0 (ττ 2 , 1 )V (τ 1 )U0 (τ 1 ,t0 )
where we have used the composition property of Ut( ,t0 ). The same positive time-ordering applies. Note that the interactions V(τi) are not in the interaction representation here. Rather we used the definition in eq. (4.28) and collected terms. τ1 For transitions between two eigenstates of H0 ( l and k): The system evolves in eigenstates of H0 during the different time periods, l V k H0 H0 with the time-dependent interactions V driving the transitions between H0 these states. The time-ordered exponential accounts for all possible H0
H0 intermediate pathways. V H0 V Also, the time evolution of conjugate wavefunction in the τ1 τ2 interaction picture is expressed as m
⎡+i t ⎤ ⎡+i t ⎤ Ut† (, t )= U † (t,t)Ut† (,t )= ex pdτ V()τ expdτ H()τ (4.34) 0 I 0 0 0 − ⎢ ∫t I ⎥ − ⎢ ∫t 0 ⎥ ⎣ 0 ⎦ ⎣ 0 ⎦
† iH (t−t0 ) or U0 = e when H0 is independent of time.
The expectation value of an operator is:
At() = ψ (t) Aψ (t)
† = ψ (tU0 ) (t,t0 ) AU (t,t0 )ψ () t0 (4.35) † † = ψ ()tU0 I U0 AU0U I ψ ()t0
= ψ I ()tAI ψ I ()t
† where AI ≡ UA0 S U0 (4.36)
Differentiating AI gives: p. 40
∂ i A = [H, A] (4.37) ∂t I 0 I
∂ −i also, ψ = Vt()ψ (4.38) ∂t I I I
Notice that the interaction representation is a partition between the Schrödinger and Heisenberg representations. Wavefunctions evolve under VI , while operators evolve under H0.
∂A ∂ −i For H = 0, V()t= H ⇒ = 0; ψ = H ψ Schrödinger 0 ∂t ∂t S S (4.39) ∂Ai ∂ψ For H = H, V()t= 0 ⇒ = [H, A]; = 0 Heisenberg 0 ∂t ∂t