# Infinite quantum wave function

1. Mar 7, 2008

### lark

Can a quantum wave function be infinite at a point? For example you could have a radially symmetric wavefunction that's infinite at the center, yet the integrated probability is 1. Is this unphysical somehow?
Laura

2. Mar 7, 2008

### clem

You are describing the Dirac delta function, the wave function of a particle with a precise position.

3. Mar 7, 2008

### Staff: Mentor

A proper wave function must have a slope that is a continuous function. I don't think this is possible at a point where the function goes to infinity. However, such a function might appear as an idealized limiting case of a sharply "peaked" wave function. For example:

4. Mar 8, 2008

### lark

I was thinking about *something* like probability density = exp(-r)/r. Something that goes infinite at the center yet has finite integral. Perfectly well-behaved except at one point. The Dirac delta function wouldn't ever appear in reality although it might as an intermediate step in one's calculations.
Laura

5. Mar 8, 2008

### ZapperZ

Staff Emeritus
What is the fourier transform of a plane wave? This will be the wavefunction in momentum space.

Zz.

6. Mar 9, 2008

### mhill

now we are talking about dirac deltas, the wavefunction

$$D^{n} \delta (x-a)$$ (derivative of delta function) has a meaning ?

from Fourier analysis, we could consider the wave fucntion above the Fourier transform of $$x^{n}$$

7. Mar 10, 2008

### peter0302

If the wave function represents probability amplitudes by definition, how could it be greater than 1 at any point?

8. Mar 10, 2008

### CPL.Luke

because its a probability density, you must integrate it to get the probability the particle is in a given range.

this situation is unphysical, it corresponds to a wavefunction with infinite expected energy.

to see this write the delta in momentum space (neglecting the various constants its e^ipx')

now use the momentum space hamiltonian for a free particle ((p^2)/2m)) and take the expectation value.

you would get an infinite result, however this is not the only reason why it is unphysical, what the delta wavefunction means in position space is a plane wave in momentum space, this plane wave in momentum space would not be normalizable and thus unphysical.

9. Mar 10, 2008

### pam

Have you heard of delta function normalization?