**Required math: calculus**

**Required physics: **Schrödinger equation

References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Section 3.3.2; Problem 3.9.

The example of a periodic function which we studied earlier had discrete eigenvalues for both the first and second derivative of the periodic variable. In particular, for the operator we found that the eigenvalues are all integers, with eigenfunctions since

This operator bears a strong resemblance to the momentum operator in one dimension, which is . However, if we try to find the eigenvalues and eigenfunctions of , we run into a bit of a problem. We try to solve, for some eigenvalue :

This has the solution

for some constant . Ordinarily, at this stage, we would impose some boundary condition on the solution to obtain acceptable values of . The problem is that we’d like to define this function over all and, if we try to do this, the function is not normalizable for any value of . At first glance, we might think that if we chose to be purely imaginary as in , it might work since we get

but of course this tends to infinity at large negative so that doesn’t work. In fact if has a non-zero imaginary part, goes to infinity at one end of its domain. So we’re restricted to looking at real values of .

In that case, is periodic and thus is still not normalizable. Thus there are no eigenfunctions of the momentum operator that lie in Hilbert space (which, remember, is the vector space of square-integrable functions).

What happens if do the normalization integral anyway? That is, we try

By using the variable transformation , we get

It’s at this point that we invoke the dodgy formula involving the Dirac delta function that we obtained a while back. Using this, we can write the integral as a delta function, and we get

This is sort of like a normalization condition, in that the integral is zero when (that is, if you believe that the integral really does evaluate to a delta function), and non-zero (infinite, in fact) if . In fact, if we take the constant to be

and use the bra-ket notation for the integral, we can write

We can also express an arbitrary function as a Fourier transform over by writing

From Plancherel’s theorem, we can invert this relation to get :

In general, hermitian operators with continuous eigenvalues don’t have normalizable eigenfunctions and have to be analyzed in this way. In particular, the hamiltonian (energy) of a system can have an entirely discrete spectrum (infinite square well or harmonic oscillator), a totally continuous spectrum (free particle, delta function barrier or finite square barrier) or a mixture of the two (delta function well or finite square well).

Pingback: Infinite square well: momentum « Physics tutorials

Pingback: Momentum space: harmonic oscillator « Physics tutorials

Pingback: Momentum space: mean position « Physics tutorials

Pingback: Position operator: eigenfunctions « Physics tutorials

Pingback: Hamiltonian matrix elements « Physics tutorials

Pingback: Sequential measurements « Physics tutorials

Pingback: Momentum space representation of finite wave function « Physics tutorials

Pingback: Momentum space: another example « Physics tutorials

Pingback: Free particle in momentum space « Physics tutorials

Pingback: Eigenfunctions of position and momentum; unit operators | Physics pages

Pingback: Path integrals in quantum mechanics | Physics pages

Pingback: Position and momentum unit operators | Physics pages

Pingback: Non-denumerable basis: position and momentum states | Physics pages

Pingback: Differential operators – matrix elements and hermiticity | Physics pages

Pingback: Postulates of quantum mechanics: momentum | Physics pages

Pingback: Parity transformations | Physics pages