Space between bound state energies in some potential.

[rar0308]

Is there a way to know qualitative information about energy spacing of bound state energy?

Infinite square well.
V=0 -a/2<x<a/2
V=∞ otherwise
Bound state energy E$\propto$ n^2
space beteween succesive energies increases at higher energy
(n+1)^2-n^2=2n+1

Harmonic Oscillator
V$\propto$x^2
E$\propto$ n
space beteween succesive energies is the same at higher energy
n+1-n=1

Coloum potential
V$\propto$x^-1
E $\propto$n^-2
space beteween succesive energies decreases at higher energy
1/(n+1)^2-1/n^2=(2n+1)/n^2(n+1)^2


For instance, How about triangular well
V(x)=|x| -∞<x<∞
Does space between successive bound energies increase or decrease at higher energy?
How can I know this?

 

[Simon Bridge]

You can compare the candidate well with similar ones you know the solutions for - that's why your lessons involve calculating energy levels for a lot of simple wells.

If the candidate well is qualitatively similar in some way then chances are the energy level spacing will be qualitatively similar ... you develop an intuition for it with practice.

A clue is the comparison with the harmonic oscillator - this has evenly spaced energy levels.
Consider: wider wells have closer-spaced energy levels than narrow wells.

i.e. your example: it is closest to the harmonic oscillator - except it get wider faster ... the HO has evenly spaced levels. Getting wider makes it get closer to the continuum faster, which should make the energy levels get closer together.

Compare infinite square well - which says the same - so the energy levels will get farther apart.

 

[rar0308]

More wider potential, More spatially spreaded wavefunction.
More spatially spreaded, Less space between bound state energies.
Is this what you mean?
Is there something like theorem which states second line of my quick reply?

When do you meet this problem first?
What book should I refer to?

 

[Simon Bridge]

You know what is meant by "an infinite square well with width L"?

In general, a potential well does not have a constant width. It's width changes with energy.
When you sketch the potential, you are sketching a graph of energy vs something. At a particular energy level, it will have a particular width. The width is measured horizontally and may be a width in space or in momentum or whatever the horizontal axis measures.

Consider the infinite square well ... the separation of the 1st two energy levels depends on the width f the well right? Check by putting L=a and L=2a and comparing.

Just to be clear: there is no "space between energy levels" - the gap there is not space, it is energy.

It is not a problem and you see the effect (wider wells have closer spaced energy levels) first when you study the infinite square well. Later on you will study "density of states" which is how many states fit into an energy interval dE and how that varies along the energy axis.

 

[rar0308]

"space between energy levels" here i used the word space to mean gap between energies.
In all potentials I solved Schroedinger Equation for, gap between bound energies gets smaller at higher energy.
But what guarantee that that would hold for potentials which I didn't solve SE for?
i.e. It might happen to be the case for some potentials, but it might not for others. How can you prove it for general potential?

 

[Simon Bridge]

"space between energy levels" here i used the word space to mean gap between energies.

That's fine - I had to check because you also used the word "space" to refer to difference in position coordinates as in: "spacial extent". It's not an uncommon misunderstanding ;)

In all potentials I solved Schroedinger Equation for, gap between bound energies gets smaller at higher energy.

Which ones have you solved the SE for then? Because that's very surprising. Show me.

The gap in question is: $\Delta E_n=\small E_{n}-E_{n-1}$ right?

Go compute that relationship for the infinite square well and show me what you've done.

 

[rar0308]

"In all potentials I solved Schroedinger Equation for, gap between bound energies gets smaller at higher energy."
Oh,I have to insert some words to above sentence. I didn't mean it.
"In all potentials I solved Schroedinger Equation for, As potential get wider/narrower, gap between bound energies gets smaller/bigger at higher energy."
Like infinite square well, square well for getting bigger case, hydrogen atoms for getting smaller case.

 

[Simon Bridge]

Oh, in that case it is pretty much guaranteed that a wider potential gives a higher density of states, yep.
The ultimate wide potential well is the free-particle potential ... what's it's density of states?
The wider the well, the closer the system is to being a free particle.