How Do You Normalize a Non-Integrable Wave Function in Quantum Mechanics?

[invisiblefrog]

I'm trying to understand what is the correct rule for the Dirac delta normalization of a non-integrable wave function, and can't seem to find any decent references. My issue is with achieving the proper dimensionality of the resulting wave function. This would be length^-1/2 for the states of a free-particle in one-dimensional space that I am considering.

Generally the normalization condition is given as
$\left\langle j | j' \right\rangle = \delta (j - j')$
where j is some kind of continuous index, but the question is what...
For example, http://physics.ucsd.edu/~emichels/FunkyQuantumConcepts.pdf goes for $\left\langle p' | p \right\rangle = \delta (p - p')$ which hardly seems satisfactory, as the units of the delta function are the inverse of those of its argument. I would want the normalization to be dimensionless, rather than equating it to something with units 1/momentum. After the calculations which logically follow from this premise, a wave function with dimensions $\hbar ^{-1/2}$ is obtained. If instead of momentum I use the wave vector (i.e. k if I am using $e ^{ikx}$), then it ends up being dimensionless, which is equally undesirable.
What condition can be used to get a usable wave function with units of length^-1/2?

 

[BruceW]

For example, http://physics.ucsd.edu/~emichels/FunkyQuantumConcepts.pdf goes for $\left\langle p' | p \right\rangle = \delta (p - p')$ which hardly seems satisfactory, as the units of the delta function are the inverse of those of its argument. I would want the normalization to be dimensionless, rather than equating it to something with units 1/momentum.

OK, why not use $\left\langle x' | x \right\rangle = \delta (x - x')$ ?

 

[dauto]

These are momentum space wave functions. They do not normalize to units length-1/2 as coordinate space wave functions do. ⟨p′|p⟩=δ(p−p) is correct.

 

[invisiblefrog]

Sorry, I should have been more clear about which functions I was trying to normalize. I am looking for a normalization constant A in $\psi(x) = Ae^{ikx}$

OK, why not use $\left\langle x' | x \right\rangle = \delta (x - x')$ ?

I don't understand how I could do this since x doesn't tell me something that distinguishes one state from another.

These are momentum space wave functions. They do not normalize to units length-1/2 as coordinate space wave functions do. ⟨p′|p⟩=δ(p−p) is correct.

The link I mentioned (page 68) defines the states it is normalizing $\left|p\right\rangle = {\psi}_p (x) = Ne^{ipx/\hbar}$ . Isn't it in coordinate space if it is a function of x?

 

[BruceW]

Sorry, I should have been more clear about which functions I was trying to normalize. I am looking for a normalization constant A in $\psi(x) = Ae^{ikx}$

hmm. you can't normalise this wavefunction though. So maybe the problem would be more clear if you used a wavefunction which can be normalised, like a wavepacket for example. (I'm still not entirely sure what your original question is, but maybe considering a wavepacket instead will help).

 

[invisiblefrog]

hmm. you can't normalise this wavefunction though. So maybe the problem would be more clear if you used a wavefunction which can be normalised, like a wavepacket for example. (I'm still not entirely sure what your original question is, but maybe considering a wavepacket instead will help).

Certainly it is impossible to normalize the function in the standard way. The question is about a special procedure called "delta function normalization" which produces a normalization constant for these non-integrable functions. The inner product of the state with itself will be infinite, equal to some multiple of a delta function. The question is what that multiple should be.

 

[dauto]

Sorry, I should have been more clear about which functions I was trying to normalize. I am looking for a normalization constant A in $\psi(x) = Ae^{ikx}$



I don't understand how I could do this since x doesn't tell me something that distinguishes one state from another.


The link I mentioned (page 68) defines the states it is normalizing $\left|p\right\rangle = {\psi}_p (x) = Ne^{ipx/\hbar}$ . Isn't it in coordinate space if it is a function of x?

The notation in the linked text is a bit sloppy in my opinion. An all too common problem with many quantum mechanical texts because, apparently, quantum mechanics isn't hard enough as is...

|p> and ψ_p(x) are not the same mathematical object even though they are both used to describe the same physical state, namely a plane wave. |p> is an eigenvector of the momentum operator. ψ_p(x) is a wavefunction. they are related to each other by <p|x> = ψ_p(x) where |x> is an eigenvector of the position operator. That explains why |p> and ψ_p(x) have different dimensions. Now, to answer your question, as pointed out by BruceW, the wave function e^(ikx) cannot be normalized in the usual way because that function is not square integrable. The integral diverges which means that plane waves cannot be represented by an element of the Hilbert space. That should not be surprising since strictly speaking plane waves occupy the whole space and have infinite energy.

 

[invisiblefrog]

OK, thanks for the help guys, I must be doing it wrong. My apologies for demanding a normalization condition which is seemingly nonexistent.