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I was wondering about the following:

If you have the time-dependent Schrödinger equation such that $$i \hbar \frac{\partial\psi(x,t)}{\partial t} = - \frac{\hbar^2}{2m} \frac{\partial^2\psi(x,t)}{\partial x^2} + V(x,t) \psi(x,t),$$

where the potential is also time dependent. What is the general strategy to solve this one? Separation of Variables or are there better techniques available? Especially if $V(x,t) = V_1(t)V_2(x)$. For example if you know the solution to $$E_n = - \frac{\hbar^2}{2m} \frac{\partial^2\psi(x,t)}{\partial x^2} + V_2(x) \psi(x),$$ does this help to find the general solution?

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2 Answers 2

up vote 9 down vote accepted

Firstly, there are a few issues with a time-dependent potential, $V(x,t)$. Namely, if we apply Noether's theorem, the conservation of energy may not apply. Specifically, if under a translation,

$$t\to t +t'$$

the Lagrangian $\mathcal{L}=T-V(x,t)$ changes by no more than a total derivative, then conservation of energy will apply, but this resricts the possible $V(x,t)$, depending on the system.


We often treat each Schrödinger equation case by case, as a certain system may lend itself to a different approach, e.g. the harmonic oscillator is easily solved by employing the formalism of creation and annihilation operators. If we consider a time-dependent potential, the equation is generally given by,

$$i\hbar\frac{\partial \psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial \mathbf{x}^2} + V(\mathbf{x},t)\psi$$

Depending on $V$, the Laplace or Fourier transform may be employed. Another approach, as mentioned by Jonas, is perturbation theory, whereby we approximate the system as a simpler system, and compute higher order approximations to the fully perturbed system.


Example

As an example, consider the case $V(x,t)=\delta(t)$, in which case the Schrödinger equation becomes,

$$i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + \delta(t)\psi$$

We can take the Fourier transform with respect to $t$, rather than $x$, to enter angular frequency space:

$$-\hbar\omega \, \Psi(\omega,x)=-\frac{\hbar^2}{2m}\Psi''(\omega,x) + \psi(0,x)$$

which, if the initial conditions are known, is a potentially simple second order differential equation, which one can then apply the inverse Fourier transform to the solution.

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ah interesting example. could you explain what an appropriate initial condition could be? (a simple example would be great) –  Xin Wang Jun 19 at 18:23
    
sorry, don't know whether I have to ping you... –  Xin Wang Jun 19 at 21:55
1  
@user180097: I honestly don't know what an appropriate initial condition would be for $\psi(0,x)$. –  JamalS Jun 20 at 5:29
    
thanks you...I will ask another question about it:-) –  Xin Wang Jun 20 at 12:22
    

I'm aware of no general recipe. If the time-dependent part of $V$ is weak, one can apply time-dependent perturbation theory (TDPT) to calculate corrections to the unperturbed, time-independent solution. This should be contained in any book on quantum mechanics. This way, one can also calculate the transition probabilities and rates. Specifically for periodic perturbations, this leads to Fermi's golden rule which can often be applied without going through the whole machinery of TDPT.

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