Harmonic Oscillator Potential Approximation

So, in summary, the potential V(x) = κ(x2-l2)2 has a stable equilibrium at x = l. Expanding the potential about this point gives V = 4κl2(x2-2lx+l2) + O(x-l)3. This can be approximated as V ≈ 1/2(8κl)(x-l)2, which is the same form as a simple harmonic oscillator potential. Therefore, the ground state energy is given by E0 = 1/2ħω, where ω = √(8κl2/m).
  • #1
thelonious
15
0

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 ~ AeC*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?
 
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  • #2
thelonious said:

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 ~ AeC*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.
$$V \cong \frac{1}{2}(8\kappa l)(x-l)^2$$
 
  • #3
So, in calculating <0|H|0>, I will have to integrate ∫(x-l)2e-mωx2/[itex]\hbar[/itex]dx from l-[itex]\epsilon[/itex] to l+[itex]\epsilon[/itex]
Right?
 
  • #4
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?
 
  • #5
So E=[itex]\hbar[/itex]ω/2, and V=1/2*kx2=1/2*8κl2(x-l)2.
ω=√(8kl2/m), so E0=[itex]\hbar[/itex]/2*√(8kl2/m)
 
  • #6
Right.
 

What is a harmonic oscillator potential approximation?

A harmonic oscillator potential approximation is a mathematical model used to describe the behavior of a particle in a potential well that is approximately parabolic in shape. It is often used in quantum mechanics to simplify calculations and provide a more accurate representation of a system.

How is a harmonic oscillator potential approximation different from a true harmonic oscillator?

A harmonic oscillator potential approximation is an approximation of a true harmonic oscillator, which is a system in which the restoring force is directly proportional to the displacement from the equilibrium position. In a true harmonic oscillator, the potential energy curve is a perfect parabola, while in a harmonic oscillator potential approximation, the curve is only approximately parabolic.

What factors can affect the accuracy of a harmonic oscillator potential approximation?

The accuracy of a harmonic oscillator potential approximation can be affected by the shape of the potential well, the strength of the restoring force, and the mass of the particle. It is also important to consider the energy levels of the system and whether they are discrete or continuous.

How is a harmonic oscillator potential approximation used in quantum mechanics?

In quantum mechanics, the harmonic oscillator potential approximation is used to solve the Schrödinger equation for systems that can be approximated as a harmonic oscillator. This allows for the calculation of energy levels and wavefunctions for these systems, providing important insights into their behavior.

What are some real-world examples of systems that can be described using a harmonic oscillator potential approximation?

Some examples of systems that can be described using a harmonic oscillator potential approximation include atomic vibrations, molecular bonds, and the motion of a mass attached to a spring. It is also often used in the study of quantum dots and nanomechanical systems.

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