# Time Evolution of States

### Notation

• $$|\alpha\rangle$$ at $$t=t_0$$, $$|\alpha,t_0\rangle$$
• $$|\alpha,t_0\rangle$$ at time $$t$$, $$|\alpha,t_0;t\rangle$$
• $$|\alpha,t_0;t\rangle = \hat{U}(t,t_0)|\alpha,t_0\rangle$$, $$\hat{U}(t,t_0)$$ is the propagator, or time evolution operator

## Deriving Propogator

• $$\hat{U}(t_0,t_0)=\mathbb{I} = \hat{U}^\dagger\hat{U}$$
• $$\hat{U}(t_0+dt,t_0) = \mathbb{I} - \frac{i}{\hbar}\hat{H}dt$$
• $$\hat{U}(t+dt,t)\hat{U}(t,t_0)=\hat{U}(t+dt,t_0)$$
• $$(\mathbb{I}-\frac{i}{\hbar}\hat{H}dt)\hat{U}(t,t_0) = \hat{U}(t,t_0)-\frac{i}{\hbar}dt\hat{H}\hat{U}(t,t_0) = \hat{U}(t+dt,t_0)$$
• $$-\frac{i}{\hbar}dt\hat{H}\hat{U}(t,t_0) = \hat{U}(t+dt,t_0)-\hat{U}(t,t_0) = \frac{\partial}{\partial t}\hat{U}(t,t_0)dt$$
• $$-\frac{i}{\hbar}\hat{H}\hat{U}(t,t_0) = \frac{\partial}{\partial t}\hat{U}(t,t_0)$$
• $$\hat{H}\hat{U}(t,t_0) = i\hbar\frac{\partial}{\partial t}\hat{U}(t,t_0)$$
• $$\hat{H}\hat{U}(t,t_0)|\alpha,t_0\rangle = i\hbar\frac{\partial}{\partial t}\hat{U}(t,t_0)|\alpha,t_0\rangle$$
• $$\hat{H}|\alpha,t_0;t\rangle = i\hbar\frac{\partial}{\partial t}|\alpha,t_0;t\rangle$$

## Types of Propagators

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

$$\hat{U}(t,t_0) = \exp\left(-\frac{i}{\hbar}\hat{H}(t-t_0)\right)$$

Assume $$\{|E_i\rangle\}$$ are eigenstates of the Hamiltonian. Then, $$|\alpha,t_0\rangle = \sum_n |E_n\rangle\langle E_n|\alpha,t_0\rangle= \sum_n c_n(0)|E_n\rangle$$. So the time translated is, $$|\alpha,t_0=0;t\rangle = \sum_n \exp\left(-\frac{i}{\hbar}E_n t\right)|E_n\rangle\langle E_n|\alpha,t_0\rangle = \sum_n \exp\left(-\frac{i}{\hbar}E_n t\right)c_n(0)|E_n\rangle = \sum_n c_n(t)|E_n\rangle$$.

### Case 2 $$\hat{H} = \hat{H}(t)$$ but $$[\hat{H}(t_1),\hat{H}(t_2)] = 0$$

$$\hat{U}(t,t_0) = \exp\left(-\frac{i}{\hbar}\int_{t_0}^t dt' \hat{H}(t')\right)$$

#### 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 $$[\hat{H}(t_1),\hat{H}(t_2)] \neq 0$$

$$\hat{U}(t,t_0) = \mathbb{I} + \sum_n \left(\frac{i}{\hbar}\right)^n\int_{t_0}^tdt_1\int_{t_0}^{t_1}dt_2\cdots\int_{t_0}^{t_{n-1}}dt_n \hat{H}(t_1)\hat{H}(t_2)\cdots \hat{H}(t_n)$$ - 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

$$\frac{d}{d t}\langle A\rangle = \frac{1}{i\hbar}\langle [A,H]\rangle + \langle \frac{\partial A}{\partial t}\rangle$$.

Proof.

$$\langle A\rangle = \langle \Psi|A|\Psi\rangle$$

$$\frac{d}{d t}\langle A\rangle = \langle \frac{\partial \Psi}{\partial t}|A|\Psi\rangle + \langle \Psi|\frac{\partial A}{\partial t}|\Psi\rangle + \langle \Psi|A|\frac{\partial \Psi}{\partial t}\rangle = -\frac{1}{i\hbar}\langle H\Psi|A|\Psi\rangle + \frac{1}{i\hbar}\langle \Psi|A|H\Psi\rangle + \langle \Psi|\frac{\partial A}{\partial t}|\Psi\rangle = -\frac{1}{i\hbar}\langle \Psi|HA|\Psi\rangle + \frac{1}{i\hbar}\langle \Psi|AH|\Psi\rangle + \langle \Psi|\frac{\partial A}{\partial t}|\Psi\rangle = \frac{1}{i\hbar}\langle [A,H]\rangle + \langle \frac{\partial A}{\partial t}\rangle$$.

### Results

#### Constants of Motion

Thus, if $$A\neq A(t)$$ then $$\frac{d}{dt}\langle A\rangle = \frac{1}{i\hbar}\langle [A,H]\rangle$$. 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 $$\frac{d}{dt}\langle B\rangle = \langle \frac{\partial B}{\partial t}\rangle$$. If we choose $$|E_\kappa\rangle$$, an eigenstate of the Hamiltonian, then $$\langle E_\kappa|B|E_\kappa\rangle|_t=\langle E_\kappa|\exp(iE_\kappa t/\hbar)B\exp(-iE_\kappa t/\hbar)|E_\kappa\rangle=\langle E_\kappa|B|E_\kappa\rangle$$. Thus, the eigenstates of this Hamiltonian are the Stationary states. With $$\frac{d}{dt}\langle E_\kappa|B|E_\kappa\rangle = 0$$ which implies that $$\langle [B,H]\rangle = 0$$ for these states.

#### Superposition of Stationary States

$$\langle B\rangle$$, $$|\alpha,t_0\rangle = \sum_n C_n(0)|E_n\rangle$$, $$|\alpha,t_0;t\rangle = \sum_n C_n(0)\exp(-iE_nt/\hbar)|E_n\rangle$$.

$$\langle B\rangle = \sum_{nm}C_m^*(0)C_n(0)\exp(iE_mt/\hbar)\exp(-iE_nt/\hbar)\langle E_m|B|E_n\rangle= \sum_{nm}C_m^*(0)C_n(0)\exp(it(E_m-E_n)/\hbar)\langle E_m|B|E_n\rangle= \sum_{nm}C_m^*(0)C_n(0)\exp(-it\omega_{nm})/\hbar)\langle E_m|B|E_n\rangle$$.

$$\omega_nm=\frac{(E_n-E_m)}{\hbar}$$ is the Bohr frequency.

#### Virial Theorem

Created: 2024-05-30 Thu 21:21

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