Time-evolution operator
The time-evolution operator $U(t,t_0)$ is defined as the operator which acts on the ket at time $t_0$ to produce the ket at some other time $t$:
- $ \ket{\Psi(t)}=\hat{U}(t,t_0)\ket{\Psi(t_0)} $
- $ \bra{\Psi(t)}=\bra{\Psi(t_0)}\hat{U}^\dag(t,t_0) $
Properties
- Unitarity
- $ \hat{U}^\dag(t,t_0)\hat{U}(t,t_0)=I $
- Identity
- $ \hat{U}(t_0,t_0)=I $
- Closure
- $ \hat{U}(t,t_0)=\hat{U}(t,t_1)\hat{U}(t_1,t_0) $
Differential equation for time-evolution operator
We drop the $t_0$ index in the time evolution operator with the convention that $t_0=0$ and write it as $U(t)$.
Using the Schrödinger equation, we get the differential equation of $U$.
\[ i\hbar\frac{\partial}{\partial t}\hat{U}(t)\ket{\Psi(0)} = \hat{\mathcal{H}}\hat{U}(t)\ket{\Psi(0)} \]
\[ \Rightarrow i\hbar\frac{\partial}{\partial t}\hat{U}(t) = \hat{\mathcal{H}}\hat{U}(t) \]
If the Hamiltonian is independent of time, the solution to the above equation is:
\[ \hat{U}(t) = e^{-\frac{it}{\hbar}\hat{\mathcal{H}}} \]
If the Hamiltonian is dependent on time, but the Hamiltonians at different times commute, then the time evolution operator can be written as:
\[ \hat{U}(t) = \exp\left( -\frac{i}{\hbar}\int_0^t\hat{\mathcal{H}}(t\rq)dt\rq \right) \]
If the Hamiltonian is dependent on time, but the Hamiltonians at different times do not commute, then the time evolution operator can be written in following form where $\mathcal{T}$ is time-ordering operator, which is sometimes known as the Dyson series.
\[ \hat{U}(t) = \mathcal{T}\exp\left( -\frac{i}{\hbar}\int_0^t\hat{\mathcal{H}}(t\rq)dt\rq \right) \]
See the wiki for more information. (Dyson series)
Dynamical pictures
Dynamical pictures (or representations) are the multiple equivalent ways to mathematically formulate the dynamics of a quantum system.
Schrödinger picture
Schrödinger picture is a method of fixing the observable and interpreting the state vector to change over time.
- Ket state: $ \ket{\Psi_S(t)}=\hat{U}(t)\ket{\Psi_S(0)} $
- Observable: constant
Heisenberg picture
Heisenberg picture is a method of fixing the state vector and interpreting that the observable changes over time.
- $ \expct{\hat{A}}_t = \brktop{\Psi(t)}{\hat{A}}{\Psi(t)} = \brktop{\Psi(0)}{\hat{U}^\dag(t)\hat{A}\hat{U}(t)}{\Psi(0)} $
- $ \hat{A}_H(t) \coloneqq \hat{U}^\dag(t)\hat{A}\hat{U}(t) $
Let’s find out then how the expectation value of an observable changes over time.
\[ \begin{align*} \frac{d}{dt} \hat{A}_H(t) &= \frac{\partial\hat{U}^\dag}{\partial t}\hat{A}\hat{U} + \hat{U}^\dag\frac{\partial\hat{A}}{\partial t}\hat{U} + \hat{U}^\dag\hat{A}\frac{\partial\hat{U}}{\partial t} \nl &= \frac{i}{\hbar}\hat{U}^\dag\hat{\mathcal{H}}\hat{A}\hat{U} + \hat{U}^\dag\frac{\partial\hat{A}}{\partial t}\hat{U} + \frac{i}{\hbar}\hat{U}^\dag\hat{A}(-\hat{\mathcal{H}})\hat{U} \nl &= \frac{i}{\hbar}\left(\hat{\mathcal{H}}\hat{A}_H-\hat{A}_H\hat{\mathcal{H}}\right) + \left(\frac{\partial\hat{A}}{\partial t}\right)_H \end{align*} \]
\[\therefore\; \boxed{ \frac{d}{dt}\hat{A}_H(t) = \frac{i}{\hbar}\com{\hat{\mathcal{H}}}{\hat{A}_H(t)} + \left(\frac{\partial\hat{A}}{\partial t}\right)_H }\]
In conclusion,
- Ket state: constant
- Observable: $ \hat{A}_H(t)=\hat{U}^\dag(t)\hat{A}\hat{U}(t) $
Interaction picture
https://en.wikipedia.org/wiki/Dynamical_pictures
Comparison of pictures
| Evolution | Picture | ||
|---|---|---|---|
| Schrödinger (S) | Heisenberg (H) | ||
| Ket state | $ \ket{\Psi_S(t)}=e^{-i\frac{\hat{\mathcal{H}}_S}{\hbar}t}\ket{\Psi_S(0)} $ | constant | $ \ket{\Psi_I(t)}=e^{i\frac{\hat{\mathcal{H}}_0}{\hbar}t}\ket{\Psi_S(t)} $ |
| Observable | constant | $ \hat{A}_H(t)=e^{i\frac{\hat{\mathcal{H}}_S}{\hbar}t}\hat{A}_S e^{-i\frac{\hat{\mathcal{H}}_S}{\hbar}t} $ | $ \hat{A}_I(t)= e^{i\frac{\hat{\mathcal{H}}_0}{\hbar}t}\hat{A}_S e^{-i\frac{\hat{\mathcal{H}}_0}{\hbar}t} $ |