Harmonic Oscillator Potential Approximation

[thelonious]

Homework Statement

A particle is in a region with the potential
V(x) = κ(x²-l²)²
What is the approximate ground state energy approximation for small oscillations about the location of the potential's stable equilibrium?

Homework Equations

ground state harmonic oscillator ~ Ae^C*x2

The Attempt at a Solution

My plan is to find the value of x for which V(x) is at stable equilibrium, then expand about that point, and the expansion should look something like the harmonic oscillator potential (1/2*mωx²). Once the potential is known, I know H = p²/2m + V, and I can use <0|H|0> to find the ground state energy, where |0> is the GS energy eigenstate for the harmonic oscillator.

I have found that the stable equilibrium of V(x) is at x = l.
Next, I expanded V(x) about this point and found:
V = 4κl²(x²-2lx+l²) + O(x-l)³

But what can I do about the 2lx term? I need it to get lost in order to get the HO potential, right?

 

[vela]

Homework Statement

A particle is in a region with the potential
V(x) = κ(x²-l²)²
What is the approximate ground state energy approximation for small oscillations about the location of the potential's stable equilibrium?

Homework Equations

ground state harmonic oscillator ~ Ae^C*x2

The Attempt at a Solution

My plan is to find the value of x for which V(x) is at stable equilibrium, then expand about that point, and the expansion should look something like the harmonic oscillator potential (1/2*mωx²). Once the potential is known, I know H = p²/2m + V, and I can use <0|H|0> to find the ground state energy, where |0> is the GS energy eigenstate for the harmonic oscillator.

I have found that the stable equilibrium of V(x) is at x = l.
Next, I expanded V(x) about this point and found:
V = 4κl²(x²-2lx+l²) + O(x-l)³

But what can I do about the 2lx term? I need it to get lost in order to get the HO potential, right?

You don't want to do anything. You want the potential in terms of powers of (x-l), which is exactly what you have.
$$V \cong \frac{1}{2}(8\kappa l)(x-l)^2$$

 

[thelonious]

So, in calculating <0|H|0>, I will have to integrate ∫(x-l)²e^{-mωx2/$\hbar$}dx from l-$\epsilon$ to l+$\epsilon$
Right?

 

[vela]

You don't need to calculate that expectation value, and no, that isn't how you'd calculate the expectation value either. The idea is you can use the approximation that you have a simple harmonic oscillator. What's the ground-state energy of a harmonic oscillator?

 

[thelonious]

So E=$\hbar$ω/2, and V=1/2*kx²=1/2*8κl²(x-l)².
ω=√(8kl²/m), so E₀=$\hbar$/2*√(8kl²/m)

 

[vela]

Right.