Bound states for a half harmonic oscillator

In summary, the conversation discusses the calculation of bound states for a system with a potential of (1/2)kx^2 for x>0 and infinity for x<0, known as a half harmonic oscillator. The question is whether to use the formula \int p(x) dx = (n - \frac{1}{4} ) h or \int p(x) dx = (n - \frac{1}{2} ) h to solve the problem. The speaker suggests using an exact solution rather than semiclassical formulae, and provides a hint that the eigenfunctions must have a certain property due to the wall at x=0. It is also mentioned that the set of eigenfunctions obtained this way must be all the
  • #1
praharmitra
311
1
We have a potential that is (1/2)kx^2 for x>0 and is infinity for x<0 ( half harmonic oscillator.

Now i want to calculate the bound states of the system for given E. My question is this:

Do we apply

1. [tex]\int p(x) dx = (n - \frac{1}{4} ) h[/tex] ( Since there is only one turning point that can have a connection formula. This is what is given in my book

2. [tex]\int p(x) dx = (n - \frac{1}{2} ) h[/tex] (This is what my teacher mentioned. I am not too sure about this.)

In both cases the integral is take over the entire classical path of the particle.
 
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  • #2
I'm not sure why you're messing around with semiclassical formulae when it's easy to solve this problem exactly ... Hint: on the right, an energy eigenfunction must be the same as an energy eigenfunction for the full oscillator (because they obey the same time-independent Schrodinger equation in this region), but the wall at x=0 means that the only allowed eigenfunctions must also have the property that ____________ (fill in the blank!). Only some of the eignefunctions of the full oscillator have this property ...
 
  • #3
Do what Avodyne says. But then you must also show that the set of eigenfunctions you get this way are all the eigenfunctions there are. You can do that by showing that any arbitrary eigenfunction for the half harmonic oscillator can be used to definine an eigenfunction for the full harmonic oscillator.
 

1. What is a bound state in a half harmonic oscillator?

A bound state in a half harmonic oscillator refers to a state where a particle is confined to a certain region due to the presence of a potential energy barrier. In this case, the potential energy is half of a harmonic oscillator potential, resulting in a half-symmetric potential.

2. How is a half harmonic oscillator different from a full harmonic oscillator?

A half harmonic oscillator potential is only present in one half of the oscillator, resulting in a half-symmetric potential. In contrast, a full harmonic oscillator potential is symmetric on both sides. This difference in potential energy leads to different energy levels and wave functions for the two systems.

3. What are the energy levels for a half harmonic oscillator?

The energy levels for a half harmonic oscillator are given by En = (n + 1/2)ħω, where n is the quantum number and ω is the angular frequency of the oscillator. These energy levels are discrete and can only take on certain values.

4. How are bound states in a half harmonic oscillator relevant in physics?

Bound states in a half harmonic oscillator are relevant in various areas of physics, particularly in quantum mechanics. They provide a simplified model for studying confinement and tunneling phenomena, and are also used to understand the properties of quantum systems with non-symmetric potentials.

5. Can a half harmonic oscillator have an unbound or free state?

No, a half harmonic oscillator can only have bound states due to the presence of the potential energy barrier. In order for a particle to have an unbound state, the potential energy must decrease to zero as the distance from the origin increases, which is not the case for a half harmonic oscillator potential.

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