EFunction and Energy of Infinite potential + Parabolic potential well

[Diomarte]

Homework Statement

Find eigenfunctions and the energy spectrum of a particle (its mass is m) in the potential well given by
V (x) = { +Infinity ; x < 0
{ (kx^2)/2 ; x > 0

Homework Equations

SEq.

The Attempt at a Solution

I think this is a combination of an infinite potential well combined with a parabolic approximation of a harmonic oscillator potential? I'm not 100% certain, and either way, I've spent a good 5+ hours working on this and haven't come up with much... I've read through Chapter 2.3 in Griffith's 2nd edition on The Harmonic Oscillator, and of course over the Ch 2.2 Infinite Square Well... I'm pretty sure I'm right when I say that the boundary conditions at x=0 require only odd functions such that ψ(x) = 0, and that ψ(x) is continuous at all points. Also if you have Griffiths, page 46 in Ch 2.3 explains with the ladder operators how to attain the different allowed energy levels of harmonic oscillator. I'm thinking that since this is essentially half of the parabolic approximation, that everything from this section would make sense, so long as you treat only the right side... That being said, I would greatly appreciate if somebody could give me a little more direction here, as I seem to be rather stuck...

 

[mark726]

Thanks!

Dear forum post author,

Thank you for your question. This is indeed a combination of an infinite potential well and a harmonic oscillator potential. The potential well is infinite on the left side (x<0) and parabolic on the right side (x>0). This type of potential is known as a "particle in a box with a spring."

To solve for the eigenfunctions and energy spectrum of this system, we can use the Schrodinger equation:

Hψ(x) = Eψ(x)

Where H is the Hamiltonian operator and E is the energy. Since this is a one-dimensional system, the Hamiltonian operator can be written as:

H = -h^2/2m * d^2/dx^2 + (kx^2)/2

We can use the ladder operators from the harmonic oscillator to solve for the energy levels. The ladder operators are defined as:

a+ = (1/√2mω) * (p - imωx)
a- = (1/√2mω) * (p + imωx)

Where p is the momentum operator and ω is the angular frequency of the harmonic oscillator, given by ω = √(k/m).

Using these operators, we can write the Hamiltonian as:

H = (a+ * a- + 1/2) * hω

We can then use the eigenvalue equation:

H |n> = En|n>

Where |n> is the nth eigenstate and En is the nth energy level.

Solving this equation, we get the energy levels as:

En = (n + 1/2) * hω

And the corresponding eigenfunctions as:

ψn(x) = A * (a+)^n * ψ0(x)

Where A is a normalization constant and ψ0(x) is the ground state wavefunction of the harmonic oscillator, given by:

ψ0(x) = (1/√π) * exp(-mωx^2/2h)

I hope this helps guide you in the right direction. If you have any further questions or need clarification, please don't hesitate to ask. Good luck with your studies!