Time Evolution of States

Notation

$| α ⟩$ at $t = t_{0}$ , $| α, t_{0} ⟩$
$| α, t_{0} ⟩$ at time $t$ , $| α, t_{0}; t ⟩$
$| α, t_{0}; t ⟩ = \hat{U} (t, t_{0}) | α, t_{0} ⟩$ , $\hat{U} (t, t_{0})$ is the propagator, or time evolution operator

Deriving Propogator

$\hat{U} (t_{0}, t_{0}) = I = {\hat{U}}^{†} \hat{U}$
$\hat{U} (t_{0} + d t, t_{0}) = I - \frac{i}{ℏ} \hat{H} d t$
$\hat{U} (t + d t, t) \hat{U} (t, t_{0}) = \hat{U} (t + d t, t_{0})$
$(I - \frac{i}{ℏ} \hat{H} d t) \hat{U} (t, t_{0}) = \hat{U} (t, t_{0}) - \frac{i}{ℏ} d t \hat{H} \hat{U} (t, t_{0}) = \hat{U} (t + d t, t_{0})$
$- \frac{i}{ℏ} d t \hat{H} \hat{U} (t, t_{0}) = \hat{U} (t + d t, t_{0}) - \hat{U} (t, t_{0}) = \frac{\partial}{\partial t} \hat{U} (t, t_{0}) d t$
$- \frac{i}{ℏ} \hat{H} \hat{U} (t, t_{0}) = \frac{\partial}{\partial t} \hat{U} (t, t_{0})$
$\hat{H} \hat{U} (t, t_{0}) = i ℏ \frac{\partial}{\partial t} \hat{U} (t, t_{0})$
$\hat{H} \hat{U} (t, t_{0}) | α, t_{0} ⟩ = i ℏ \frac{\partial}{\partial t} \hat{U} (t, t_{0}) | α, t_{0} ⟩$
$\hat{H} | α, t_{0}; t ⟩ = i ℏ \frac{\partial}{\partial t} | α, t_{0}; t ⟩$

Types of Propagators

Case 1 $\hat{H} \neq \hat{H} (t)$

$\hat{U} (t, t_{0}) = \exp (- \frac{i}{ℏ} \hat{H} (t - t_{0}))$

Assume ${| E_{i} ⟩}$ are eigenstates of the Hamiltonian. Then, $| α, t_{0} ⟩ = \sum_{n} | E_{n} ⟩ ⟨ E_{n} | α, t_{0} ⟩ = \sum_{n} c_{n} (0) | E_{n} ⟩$ . So the time translated is, $| α, t_{0} = 0; t ⟩ = \sum_{n} \exp (- \frac{i}{ℏ} E_{n} t) | E_{n} ⟩ ⟨ E_{n} | α, t_{0} ⟩ = \sum_{n} \exp (- \frac{i}{ℏ} E_{n} t) c_{n} (0) | E_{n} ⟩ = \sum_{n} c_{n} (t) | E_{n} ⟩$ .

Case 2 $\hat{H} = \hat{H} (t)$ but $[\hat{H} (t_{1}), \hat{H} (t_{2})] = 0$

$\hat{U} (t, t_{0}) = \exp (- \frac{i}{ℏ} \int_{t_{0}}^{t} d t^{'} \hat{H} (t^{'}))$

Example

Time varying magnetic field, but the magnetic field strength is the only thing that is varying

Case 3 $\hat{H} = \hat{H} (t)$ but $[\hat{H} (t_{1}), \hat{H} (t_{2})] \neq 0$

$\hat{U} (t, t_{0}) = I + \sum_{n} {(\frac{i}{ℏ})}^{n} \int_{t_{0}}^{t} d t_{1} \int_{t_{0}}^{t_{1}} d t_{2} \dots \int_{t_{0}}^{t_{n - 1}} d t_{n} \hat{H} (t_{1}) \hat{H} (t_{2}) \dots \hat{H} (t_{n})$ - Dyson series

Example

Time varying magnetic field, but the spin operator is also changing in time (such that the eigenbasis is shifting) (i.e. $S_{z} \to S_{x}$ )

Time Evolution of Expectation Values

Ehrenfest’s Theorem

$\frac{d}{d t} ⟨ A ⟩ = \frac{1}{i ℏ} ⟨ [A, H] ⟩ + ⟨ \frac{\partial A}{\partial t} ⟩$ .

Proof.

$⟨ A ⟩ = ⟨ Ψ | A | Ψ ⟩$

$\frac{d}{d t} ⟨ A ⟩ = ⟨ \frac{\partial Ψ}{\partial t} | A | Ψ ⟩ + ⟨ Ψ | \frac{\partial A}{\partial t} | Ψ ⟩ + ⟨ Ψ | A | \frac{\partial Ψ}{\partial t} ⟩ = - \frac{1}{i ℏ} ⟨ H Ψ | A | Ψ ⟩ + \frac{1}{i ℏ} ⟨ Ψ | A | H Ψ ⟩ + ⟨ Ψ | \frac{\partial A}{\partial t} | Ψ ⟩ = - \frac{1}{i ℏ} ⟨ Ψ | H A | Ψ ⟩ + \frac{1}{i ℏ} ⟨ Ψ | A H | Ψ ⟩ + ⟨ Ψ | \frac{\partial A}{\partial t} | Ψ ⟩ = \frac{1}{i ℏ} ⟨ [A, H] ⟩ + ⟨ \frac{\partial A}{\partial t} ⟩$ .

Results

Constants of Motion

Thus, if $A \neq A (t)$ then $\frac{d}{d t} ⟨ A ⟩ = \frac{1}{i ℏ} ⟨ [A, H] ⟩$ . Further, if they commute then the expectation value of $A$ is totally time independent, A is a constant of the motion.

If $A$ is the Hamiltonian and $H \neq H (t)$ then $H$ is a constant of the motion.

Stationary State

If $B$ does not commute with the Hamiltonian then $\frac{d}{d t} ⟨ B ⟩ = ⟨ \frac{\partial B}{\partial t} ⟩$ . If we choose $| E_{κ} ⟩$ , an eigenstate of the Hamiltonian, then $⟨ E_{κ} | B | E_{κ} ⟩ |_{t} = ⟨ E_{κ} | \exp (i E_{κ} t / ℏ) B \exp (- i E_{κ} t / ℏ) | E_{κ} ⟩ = ⟨ E_{κ} | B | E_{κ} ⟩$ . Thus, the eigenstates of this Hamiltonian are the Stationary states. With $\frac{d}{d t} ⟨ E_{κ} | B | E_{κ} ⟩ = 0$ which implies that $⟨ [B, H] ⟩ = 0$ for these states.

Superposition of Stationary States

$⟨ B ⟩$ , $| α, t_{0} ⟩ = \sum_{n} C_{n} (0) | E_{n} ⟩$ , $| α, t_{0}; t ⟩ = \sum_{n} C_{n} (0) \exp (- i E_{n} t / ℏ) | E_{n} ⟩$ .

$⟨ B ⟩ = \sum_{n m} C_{m}^{*} (0) C_{n} (0) \exp (i E_{m} t / ℏ) \exp (- i E_{n} t / ℏ) ⟨ E_{m} | B | E_{n} ⟩ = \sum_{n m} C_{m}^{*} (0) C_{n} (0) \exp (i t (E_{m} - E_{n}) / ℏ) ⟨ E_{m} | B | E_{n} ⟩ = \sum_{n m} C_{m}^{*} (0) C_{n} (0) \exp (- i t ω_{n m}) / ℏ) ⟨ E_{m} | B | E_{n} ⟩$ .

$ω_{n} m = \frac{(E_{n} - E_{m})}{ℏ}$ is the Bohr frequency.

Time Evolution of States

Notation

Deriving Propogator

Types of Propagators

Case 1 H^≠H^(t)

Case 2 H^=H^(t) but [H^(t1),H^(t2)]=0

Example

Case 3 H^=H^(t) but [H^(t1),H^(t2)]≠0

Example

Time Evolution of Expectation Values

Ehrenfest’s Theorem

Results

Constants of Motion

Stationary State

Superposition of Stationary States

Virial Theorem

Case 1 $\hat{H} \neq \hat{H} (t)$

Case 2 $\hat{H} = \hat{H} (t)$ but $[\hat{H} (t_{1}), \hat{H} (t_{2})] = 0$

Case 3 $\hat{H} = \hat{H} (t)$ but $[\hat{H} (t_{1}), \hat{H} (t_{2})] \neq 0$