Why is there only odd eigenfunctions for a 1/2 harmonic oscillator

In summary, for a 1/2 harmonic oscillator with a potential V(x) that does not equal infinity in the positive x direction, there are only odd eigenfunctions. This is because the "ground state" wave function is 0 at x=0, where V(x) is infinity, and the even solutions do not satisfy this boundary condition. Therefore, there are no even eigenfunctions and energy levels for n=2,4,6, etc.
  • #1
thegirl
41
1
Hi,

why there is only odd eigenfunctions for a 1/2 harmonic oscillator where V(x) does not equal infinity in the +ve x direction but for x<0 V(x) = infinity.

I understand that the "ground state" wave function would be 0 as when x is 0 V(x) is infinity and therefore the wavefunction is 0, and therefore the ground state energy level for the half harmonic oscillator is 3(h bar omega)/2.

I don't get why there wouldn't be even eigenfunctions and energy levels for n=2,4,6 etc.
 
Physics news on Phys.org
  • #2
thegirl said:
I don't get why there wouldn't be even eigenfunctions and energy levels for n=2,4,6 etc.

The infinite potential at 0 implies that ##\psi(0)=0##. The even solutions are all non-zero at that point, so they do not satisfy that boundary condition.
 
  • Like
Likes thegirl
  • #3
Thank You!
 

Related to Why is there only odd eigenfunctions for a 1/2 harmonic oscillator

1. Why are there only odd eigenfunctions for a 1/2 harmonic oscillator?

The reason for this is because the potential energy function for a 1/2 harmonic oscillator is an odd function. This means that when the potential energy is graphed as a function of displacement, it is symmetrical about the origin and decreases in magnitude as the displacement increases. This odd potential energy function results in the wavefunction of the system being an odd function as well, leading to only odd eigenfunctions being possible.

2. What does it mean for a potential energy function to be an odd function?

An odd function is a mathematical function that satisfies the property f(-x) = -f(x) for all values of x. This means that the function is symmetrical about the origin and has opposite values for equal distances from the origin. In the context of a 1/2 harmonic oscillator, this means that the potential energy decreases in magnitude as the displacement from the origin increases in either direction.

3. Can a 1/2 harmonic oscillator have even eigenfunctions?

No, a 1/2 harmonic oscillator can only have odd eigenfunctions due to the nature of its potential energy function. Since the potential energy is an odd function, the wavefunction must also be an odd function in order to satisfy the Schrödinger equation. Therefore, only odd eigenfunctions are possible for a 1/2 harmonic oscillator.

4. How do odd eigenfunctions affect the energy levels of a 1/2 harmonic oscillator?

The energy levels of a 1/2 harmonic oscillator are determined by the eigenfunctions of the Hamiltonian operator, which is based on the potential energy function. Since only odd eigenfunctions are possible for a 1/2 harmonic oscillator, the energy levels will be limited to half-integer values. This is in contrast to a full harmonic oscillator, which has both even and odd eigenfunctions and therefore has integer energy levels.

5. Are there any other systems where only odd eigenfunctions are possible?

Yes, there are other physical systems that have potential energy functions that are odd and therefore only allow for odd eigenfunctions. One example is the infinite square well potential, where the potential energy is equal to zero within a certain range and infinite outside of that range. This results in an odd potential energy function and only odd eigenfunctions are possible for this system as well.

Similar threads

I Particle on a cylinder with harmonic oscillator along z-axis
    • Quantum Physics
Replies
2
Views
760
I Doubt on Morse potential and harmonic oscillator
    • Quantum Physics
Replies
2
Views
1K
I Translating the harmonic oscillator
    • Quantum Physics
Replies
2
Views
887
A Angular momentum uncertainty principle and the particle on a ring
    • Quantum Physics
Replies
3
Views
533
I Qualitative plots of harmonic oscillator wave function
    • Quantum Physics
Replies
3
Views
1K
A Does there exist a proof for these conjectures?
    • Quantum Physics
Replies
2
Views
1K
A The half harmonic oscillator's ground state wave function
    • Quantum Physics
Replies
2
Views
1K
I Question about the quantum harmonic oscillator
    • Quantum Physics
Replies
7
Views
2K
I Quantum Harmonic Oscillator (QHO)
    • Quantum Physics
Replies
5
Views
3K
A Feynman propagator for a simple harmonic oscillator
    • Quantum Physics
Replies
9
Views
3K

Hot Threads

Recent Insights

Back
Top