The charged harmonic oscillator in the presence of a uniform...

The charged harmonic oscillator in the presence of a uniform electric field. The Hamiltonian that describes a particle of mass m and charge q subject to a harmonic oscillator potential in the presence of a uniform electric field E (in the x direction), is given by H = frac{p^2}{2m} + frac{1}{2} m omega^2 x^2 - qmathcal{E}x. Calculate the eigenfunctions and eigenvalues of the Hamiltonian of the system.

Transcript

00:01 All right, so we have a quantum harmonic oscillator with the addition of an external electric field.

00:09 And so our hamiltonian looks like this, p squared over 2m plus 1 1⁄2m of omega squared x squared.

00:19 So that's the normal quantum harmonic oscillator part.

00:21 And then we have minus qe times x.

00:27 And so we want to know what the new eigenfunction.

00:31 And eigenvalues of this hamiltonian are.

00:33 So there's actually a trick.

00:35 You may be familiar with it.

00:37 But if i complete the square of this term, i don't know why this happens.

00:42 If i complete the square of this term, i can actually write this single term as a square, and then we'll have a constant term outside of this.

00:51 So to show you what i mean, this can be rewritten as p squared over 2m, so that part doesn't change, plus 1⁄2m.

01:01 Omega squared times x minus q e over m omega squared squared and then we'll have a constant term which is minus q squared e squared over two m omega squared so i believe i have all that written down in my notes yeah so if we do that if you complete this square you'll notice that you know all the terms cancel out.

01:33 The reason this is useful is now we can define a new variable.

01:37 Just call it u, which is equal to x minus qe over m omega squared.

01:47 And the hamiltonian in terms of this variable u, well, the p squared over 2m part will actually look the same because we're just shifting x by a constant.

01:55 And so we can take derivatives with respect to you the same way we do with respect to x.

02:00 So the p squared over 2m, the kinetic turn doesn't change, and this becomes plus 1 half m omega squared, u squared, and then we have this constant term, minus q squared, e squared over 2m omega squared.

02:16 So this is just the hamiltonian of our ordinary harmonic oscillator that we already know, plus a constant term, which is not actually going to change the eigen functions.

02:27 So the eigen functions are going to be exactly the same as the standard.

02:31 Standard ones just with this one substitution.

02:34 So everywhere you see an x in your previous eigenfunction, you just replace it with x minus qe over m omega squared.

02:41 So for instance, psi of n in terms of x is going to be like one over, you know, we have all these coefficients and normalization terms...

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