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.