# Is the Heisenberg Picture Better for a Time-Dependent Hamiltonian?

• Mayan Fung
In summary, the conversation discusses using a completing square in the Hamiltonian to construct eigenfunctions and energy eigenvalues, and then using the Heisenberg picture to evaluate the time evolution for the annihilation-creation operators and calculate the overlap with the time-dependent eigenvectors of the Hamiltonian. This approach may provide a solution to the problem at hand.
Mayan Fung
Homework Statement
Solve the analytically the time-dependent Schrodinger equation with Hamiltonian:
$$\hat{H} = \frac{\hat{p}^2}{2} + \frac{\hat{q}^2}{2} + \alpha(t) \hat{q}$$
where ##\alpha(t) = 1-e^{-t}## for ##t \leq 0## and 0 for ##t <0##. With initial wave function in coordinate space:
$$\Psi(q, t= 0) = Nexp(-\frac{q^2}{4})$$
Relevant Equations
\begin{align}
E_n &= \hbar(n+\frac{1}{2})\\
|\Psi(t)> &= exp(-i\frac{\hat{H}}{\hbar}t)|\Psi(0)>
\end{align}
What I have tried is a completing square in the Hamiltonian so that

$$\hat{H} = \frac{\hat{p}^2}{2} + \frac{(\hat{q}+\alpha(t))^2}{2} - \frac{(\alpha(t))^2}{2}$$

I treat ##t## is just a parameter and then I can construct the eigenfunctions and the energy eigenvalues by just referring to a standard harmonic oscillator solution by shifting some constants. And then I used

\begin{align}
|\Psi(t)> &= exp(-i\frac{\hat{H}}{\hbar}t)|\Psi(0)> \\
|\Psi(t)> &= \sum_{n=0}^\infty exp(-i\frac{\hat{H}}{\hbar}t) |n><n|\Psi(0)> \\
|\Psi(t)> &= \sum_{n=0}^\infty exp(-i\frac{E_n}{\hbar}t) <n|\Psi(0)> |n>\\
\end{align}
where ## <n|\Psi(0)> = \int \Psi_n^*(t)\Psi(t=0) \, dx##

Because of the ##\alpha(t) = 1-e^{-t}##, I can't compute ## \int \Psi_n^*(t)\Psi(t=0) \, dx##. I think the problem can be more easily solved because the initial wave function ##\Psi(q, t= 0) = Nexp(-\frac{q^2}{4})## looks like that it is deliberately chosen. I am also not sure if I can extend the harmonic oscillator solution for a time varying constant.

Hint: You cannot do this so easily in this way, because the Hamiltonian is explicitly time-dependent. Also what you write as ##E_n## are the energy eigenvalues of the harmonic oscillator Hamiltonian without the extra time-dependent term.

Mayan Fung
vanhees71 said:
Hint: You cannot do this so easily in this way, because the Hamiltonian is explicitly time-dependent. Also what you write as ##E_n## are the energy eigenvalues of the harmonic oscillator Hamiltonian without the extra time-dependent term.

I tried to follow the steps suggested in this notes:

https://ocw.mit.edu/courses/chemist...pring-2009/lecture-notes/MIT5_74s09_lec02.pdf

By writing
$$|\Psi(t)>= \sum_{n=0} c_n(t) |n>$$, where ##|n>## is the harmonic oscillator wave function. And I still found that the final expression for ##c_n## doen't look analytically solvable.

If I remember right, the trick is to use the Heisenberg picture and evaluate the time evolution for the annihilation-creation (or ladder) operators and then calculate the overlap of the time-dependent (!) eigenvectors of ##\hat{H}_0## with the state, which by definition is time-independent in the Heisenberg picture and in your case given as a pure state represented by the wave function ##\exp(-q^2/4)##.

dRic2 and Mayan Fung
vanhees71 said:
If I remember right, the trick is to use the Heisenberg picture and evaluate the time evolution for the annihilation-creation (or ladder) operators and then calculate the overlap of the time-dependent (!) eigenvectors of ##\hat{H}_0## with the state, which by definition is time-independent in the Heisenberg picture and in your case given as a pure state represented by the wave function ##\exp(-q^2/4)##.

This is a clever approach. Thanks! I would have a try

## 1. What is a time dependent Hamiltonian?

A time dependent Hamiltonian is a mathematical operator that represents the total energy of a system, taking into account both its kinetic and potential energy, as well as any external forces acting on the system. This operator is used in quantum mechanics to describe the evolution of a system over time.

## 2. How is the time dependent Hamiltonian different from the time independent Hamiltonian?

The time dependent Hamiltonian takes into account the changes in a system over time, while the time independent Hamiltonian assumes that the system's properties and forces acting on it are constant. The time dependent Hamiltonian is more complex and requires more advanced mathematical techniques to solve.

## 3. What techniques are used to solve time dependent Hamiltonians?

There are several techniques used to solve time dependent Hamiltonians, including the time-dependent perturbation theory, the time-dependent variational principle, and numerical methods such as the time-evolution method and the split-operator method. Each technique has its own advantages and limitations, and the choice of method depends on the specific problem being solved.

## 4. Can time dependent Hamiltonians be solved analytically?

In most cases, time dependent Hamiltonians cannot be solved analytically, meaning that there is no closed-form solution. This is due to the complexity of the equations involved and the fact that they often involve time-dependent variables. Instead, numerical methods are typically used to approximate the solution.

## 5. How are time dependent Hamiltonians used in real-world applications?

Time dependent Hamiltonians are used in a wide range of real-world applications, including quantum mechanics, molecular dynamics, and chemical reactions. They are also used in fields such as astrophysics and engineering to model and predict the behavior of complex systems over time. Understanding and solving time dependent Hamiltonians is crucial for making accurate predictions and advancements in these fields.

Replies
24
Views
1K
Replies
10
Views
1K
Replies
1
Views
942
Replies
16
Views
997
Replies
1
Views
2K
Replies
3
Views
1K
Replies
1
Views
2K
Replies
2
Views
3K
Replies
1
Views
2K
Replies
2
Views
383