# Time-Dependent Potentials in Quantum Mechanics

A potential that depends on time is usually solved using the time dependent perturbation theory in standard undergraduate textbooks in quantum mechanics. The reason usually mentioned is that time dependent potentials cannot be solved using separation of variables.

Is this the only reason why it has to be solved using time dependent perturbation theory (Just because we cannot use methods of separation of variables)?

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There are several methods, and several reasons.

### Sepration of variables is not

The method of "separation of variables" is a misnomer in this exact context. What separation of variables really means in quantum mechanics is that you solve the eigenvalue problem for the linear operator of the Hamiltonian, and this gives the general solution by superposition. It can be formally done by a separation of variables, but the mathematical classification of the eigenfunction/eigenvalue technique is under linear algebra and functional analysis. Separation of variables is a formal trick that works for some nonlinear equations too, to find special solutions, but in the linear algebra context, it's a deep theory, not a formal trick, so it is not good to conflate the two methods.

When you have the equation

$$\partial_t \psi = A\psi$$

where A is a linear operator, the eigenvalue problem gives a basis where the equation is solved (for all diagonalizable A), because the coefficients of the eigenbasis just move according to the eigenvalue.

The method of separation of variables is only significant and interesting (other than for being equivalent to eigenbasis) for nonlinear equations, where the interpretation above doesn't exist, but you still can sometimes find product solutions anyway. For example, for the equation

$${(\partial_x\phi)^2\over \phi} + (x^2+a^2)\partial_x^2 \phi = \partial_t^2 \phi$$

Which reduces by separation of variables, but this reduction has no linear eigenvalue interpretation.

### Exact solutions

You don't have to use time-dependent perturbation theory, you can find an exact solution, like in the case of a particle in the time-dependent constant force field

$$V(t,x) = F(t)x$$

Which can be solved by an appropriate time-dependent boost of the SE

Or for a particle in certain time-dependent quadratic potential

$$V(x,t) = A(t) x^2$$

Which can be solved by path integrals (at least formally, and for certain choices of A(t)).

You can also exactly solve

$$V(x,t)= C(t)\delta(x)$$ $$V(x,t)= A(t) x$$ $$V(x,t)= B(t) x^2$$ $$V(x,t) = A(t) x + B(t) x^2$$

And if you are fastidious, you can solve the step potential with a time dependent step, superpositions of time-dependent delta functions.

### Moving hard wall

I will give the explicit solution of one of these: the moving hard wall. A Schrodinger wave with mass m and momentum k to the right hits a hard wall (an infinite potential step) moving to the left with velocity v. What happens?

The solution is a superposition of incoming and reflected wave where the reflected wave has a momentum -k+2mv and the superposition vanishes in phase at the location of the wall. This is by Galilean invariance--- the potential reduces to hard step when you galilean boost to a frame moving along with the wall.

The general solution is the superposition of these reflected waves, and is nothing more than the boosted solution to the non-moving problem.

The constant linear potential is solved by transforming to a frame that boosts by a velocity that increases linearly with time, which makes the SE free. If the boost parameter is made an arbitrary function of time, you solve the linear potential.

You can also exactly solve sudden change problems, where the potential suddenly goes from one form to another (this is the well known example of the sudden approximation), and adiabatic problems (where the potential varies very slowly in time). The adiabatic case is most interesting, because it takes eigenstates smoothly to eigenstates, except when there are collisions of eigenvalues.

### Numerical methods

Finally, you can solve the SE numerically. You place the SE on a lattice, with lattice x spacing $\epsilon$, and lattice time spacing $\epsilon^2$ (this is very important in a naive discretization, otherwise you have stability problems, see a reference on stiff equations).

You replace the Laplacian with the sum of the neighbors minus the center value, and you discretize the potential. Then the solution is a stadard numerical problem.

### Time dependent perturbation theory is the discrete path integral

The time-dependent perturbation series is just the form that the Feynman path integral takes for a discrete state space, like the eigenvalues of an already solved Hamiltonian. As such, it is just a formal solution of the problem, a formulation in an equivalent language, with truncations corresponding to each order or perturbations, which are equivalent to pulling down powers of the potential in an expansion of a particle path integral scattering.

To see this, consider the steps of deriving the path integral: you transform to a momentum basis, and push forward in time by free propagation, then into a position basis, and push forward in time by a phase for the potential. For discrete energy levels, the time dependent perturbation theory pushes forward in time using the $H_0$ Hamiltonian, then pushes forward with the interaction Hamiltonian.

The equivalence between time dependent perturbation theory and the discrete state path integral is (implicitly) given by Feynman in an obscure matheamtical paper from the early 1950s, where he develops an ideosyncratic notation for time dependent perturbation theory. It doesn't add anything to the formalism, it is just a nice point of view.

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