Prove the nth energy eigenfunction has n-1 zeros

[Happiness]

How do we show that $\psi_n(x)$ has $(n-1)$ zeros for all $n\in Z^+$?

Assuming $\psi_k(x)$ has $(k-1)$ zeros for some $k\in Z^+$, by oscillation theorem, we can only get $\psi_{k+1}(x)$ has $\geq k$ zeros.

Also, how do we show that $\psi_1(x)$, the eigenfunction corresponding to $E_1$, has no zero? Oscillation theorem may permit $\psi_1(x)$ to have 3 zeros, $\psi_2(x)$ to have 5 zeros, $\psi_3(x)$ to have 20 zeros, etc.

 

[DrClaude]

I've never heard of the oscillation theorem. Could you post the text of problem 3.9?

Edit: I just saw your other thread, where the problem is stated.

 

[stevendaryl]

Here is a paper that I found giving an argument for the nth eigenfunction having n-1 nodes:
http://arxiv.org/pdf/quant-ph/0702260.pdf

His argument is a little hard to follow, but it is short.

 

[DrClaude]

@Happiness, could you please say which book this is from? I would like to look at this in more detail.

 

[Happiness]

@DrClaude, it's Quantum Mechanics by B H Bransden and C J Joachain, 2nd edition, page 119.

 

[DrDu]

Landau Lifshitz, Vol 3 also has a good discussion.

 

[Happiness]

Here is a paper that I found giving an argument for the nth eigenfunction having n-1 nodes:
http://arxiv.org/pdf/quant-ph/0702260.pdf

His argument is a little hard to follow, but it is short.

Does he mean "As we separate the walls, N number of the wave functions ($\psi_1, \psi_2, ..., \psi_N$) go to zero at $x=\infty$ and at $x=-\infty$, and the positive energy states ($\psi_{N+1}$ onwards) become the continuum spectrum"?

 

[Happiness]

View attachment 93256

Does he mean "As we separate the walls, N number of the wave functions ($\psi_1, \psi_2, ..., \psi_N$) go to zero at $x=\infty$ and at $x=-\infty$, and the positive energy states ($\psi_{N+1}$ onwards) become the continuum spectrum"?

Now I guess he mean "As we separate the walls, the wave functions for n>N ($\psi_{N+1}$ onwards) become the everywhere-zero function" since all these wave functions have a finite number of zeros but the potential can only support an $N$ number of bound states (which have a finite number of zeros) and the other admissible wave functions are positive-energy states and thus have infinite number of zeros.

So I guess he should have said "all of these wave functions for n>N ..." instead of "some of the wave functions ..." .