Sagging Bottom Quantum Box and Pertubation

In summary, the conversation discusses a particle in a one-dimensional box potential with a sagging bottom. The potential is represented by v(x) = -V_0sin(\pi x/L) for 0 \leq x \leq L and infinity outside of this range (x > L, x < 0). The first part of the conversation involves sketching the potential as a function of x. The second part talks about treating this potential as a perturbation of a box with a straight bottom and finding the perturbation potential \Delta V(x). The third part involves calculating the energy shift for the particle in the nth stationary state to first order in the perturbation. The conversation also includes equations for an inharmonic oscillator and
  • #1
TFM
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Homework Statement



Consider a particle in a one-dimensional “box” with sagging bottom

[tex] v(x) = -V_0sin(\pi x/L) for 0 \leq x \leq L[/tex]

infinity outside of thius (x > L, x < 0)

a)

Sketch the potential as a function of x.

b)

For small [tex]V_0[/tex] this potential can be considered as a small perturbation of a “box” with a straight bottom, for which we have already solved the Schrodinger equation. What is the perturbation potential [tex]\Delta[/tex]V (x)?

c)

Calculate the energy shift due to the sagging for the particle in the nth stationary state to first order in the perturbation.

Homework Equations





The Attempt at a Solution



I have completed the first section with a graph as attached. I am not sure on the second part. I know for a inharmonic oscillator,

[tex] v(x) = V_0(x) + \lambda x^4 [/tex]

where [tex]\lambda x^4 [/tex] is the [tex] \Delta v [/tex]

But I am not sure what to do in this question.

Can anyone offer any advice?

Many Thanks

TFM
 

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  • #2
Okay, I feel that this may too be very useful for this problem

[tex] V(x) = V_0(x) + \Delta V(x) [/tex]

rouble is, this is a sum, where as the V(x) for this question appears to be the product, snce it is [tex] -V_0 * sin(\pi x /L) [/tex]

Also, may be useful, the stationary Schrödinger Equation,

[tex] E_n\phi_n(x) = -\frac{\hbar^2}{2m} + (v_0(x) + \Delta V(x))\phi_n(x) [/tex]

Is this useful?

Any suggestions?

The Ferry Man
 

1. What is the Sagging Bottom Quantum Box and Perturbation problem?

The Sagging Bottom Quantum Box and Perturbation problem is a theoretical physics problem that involves studying the behavior of a quantum particle confined to a box with a perturbation, or small disturbance, in the potential energy. This problem is often used to model real-world systems, such as atoms in a crystal lattice, and serves as a fundamental example in quantum mechanics.

2. What is the significance of the "sagging bottom" in this problem?

The "sagging bottom" refers to the shape of the potential energy in the quantum box, which is typically a square well. In this problem, the bottom of the well is lowered in the center, creating a "sagging" appearance. This perturbation allows for the study of the effects of a non-uniform potential on the behavior of a quantum particle.

3. How is the Sagging Bottom Quantum Box and Perturbation problem solved?

The problem is solved by applying mathematical techniques from quantum mechanics, such as the Schrödinger equation and perturbation theory. The solution involves finding the energy eigenvalues and corresponding wavefunctions for the perturbed potential, which describe the possible states of the quantum particle in the box.

4. What are the applications of studying the Sagging Bottom Quantum Box and Perturbation problem?

Studying this problem can provide insights into the behavior of quantum systems with non-uniform potentials, which can be found in various physical systems. It also serves as a simple and fundamental example for students learning about quantum mechanics and its mathematical tools.

5. Are there any limitations to the Sagging Bottom Quantum Box and Perturbation problem?

As with any theoretical model, there are limitations to the Sagging Bottom Quantum Box and Perturbation problem. It assumes a one-dimensional system and does not take into account factors such as temperature and external forces. Additionally, it is a simplified model and may not accurately represent more complex systems.

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