Solving Schrödinger's Equation with a Smooth Potential Wall: A Detailed Guide

In summary, the conversation was about a specific exercise in the Landau-Lifshitz book on non-relativistic quantum mechanics. The exercise involved solving a Schrödinger's equation for a smooth potential wall and finding the transmission coefficient. The conversation focused on a particular equation and how one person was having trouble getting the same result as Landau. After some discussion, it was discovered that the person had made a simple algebra error that caused the discrepancy.
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paul159753
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TL;DR Summary
I'm looking for help for an exercise of the chapter III. Schrödinger's equation, §25 The transmission coefficient (problem 3).
Hello everyone,

I'm looking for help for the problem 3 of the chapter III. Schrödinger's equation, §25 The transmission coefficient of the Volume 3 of the Landau-Lifshitz book (non-relativistic QM).

In this exercise Landau considers a smooth potential wall $$\frac{U_0}{1 + \exp{\left(-\alpha x \right)}}.$$ We search solutions for ##E > U_0##.
So we have to solve $$\frac{d^2}{dx^2}\Psi + \frac{2m}{\hbar^2}\left(E - \frac{U_0}{1 + \exp{\left(-\alpha x \right)}}\right)\Psi = 0.$$ We do the change of variable ##\xi = -\exp{\left(-\alpha x \right)}##. We write the derivative operator $$\frac{d^2}{dx^2}= \left(\frac{d\xi}{dx}\frac{d}{d\xi}\right)^2 = \alpha^2 \xi \frac{d}{d\xi} + \alpha^2 \xi ^2 \frac{d^2}{d\xi^2}.$$ Noting ##k_2^2 = 2m(E-U_0)/\hbar^2,\ k_1^2 = 2mE/\hbar^2## we can rewrite the equation $$\frac{d^2}{dx^2}\Psi + \left(\frac{k_2^2 - \xi k_1 ^2}{1 - \xi}\right)\Psi = 0.$$ We look for solutions of the form ##\Psi = \xi ^{-ik_2/\alpha} \omega(\xi)##. Multiplying the equation by ##1-\xi## and dividing by ##\alpha^2 \xi \xi ^{-ik_2/\alpha}## we get the following equation for ##\omega(\xi)## : $$ \xi(1-\xi)\frac{d^2}{d\xi^2}\omega(\xi) + (1-\xi)\left(1 - 2ik_2 / \alpha\right)\frac{d}{d\xi}\omega(\xi) + \left((k_2^2 - k_1^2)/\alpha^2 - ik_2/\alpha \frac{(1-\xi)}{\xi} \right)\omega(\xi) = 0.$$
My problem is that Landau found $$\xi(1-\xi)\frac{d^2}{d\xi^2}\omega(\xi) + (1-\xi)\left(1 - 2ik_2 / \alpha\right)\frac{d}{d\xi}\omega(\xi) + \left((k_2^2 - k_1^2)/\alpha^2 \right)\omega(\xi) = 0, $$ so the same thing but without the ##- ik_2/\alpha \frac{(1-\xi)}{\xi}\omega(\xi)## term. So he gets an hypergeometric equation and I don't. I don't know where my mistakes are.

Any help would be much appreciated.

Thanks !
 
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paul159753 said:
Multiplying the equation by ##1-\xi## and dividing by ##\alpha^2 \xi \xi ^{-ik_2/\alpha}## we get the following equation for ##\omega(\xi)## : $$ \xi(1-\xi)\frac{d^2}{d\xi^2}\omega(\xi) + (1-\xi)\left(1 - 2ik_2 / \alpha\right)\frac{d}{d\xi}\omega(\xi) + \left((k_2^2 - k_1^2)/\alpha^2 - ik_2/\alpha \frac{(1-\xi)}{\xi} \right)\omega(\xi) = 0.$$
My problem is that Landau found $$\xi(1-\xi)\frac{d^2}{d\xi^2}\omega(\xi) + (1-\xi)\left(1 - 2ik_2 / \alpha\right)\frac{d}{d\xi}\omega(\xi) + \left((k_2^2 - k_1^2)/\alpha^2 \right)\omega(\xi) = 0, $$ so the same thing but without the ##- ik_2/\alpha \frac{(1-\xi)}{\xi}\omega(\xi)## term. So he gets an hypergeometric equation and I don't. I don't know where my mistakes are.

Any help would be much appreciated.

Thanks !
When you expand the Schrodinger equation out, you should get cancellation of all the terms proportional to ##i\omega(\xi)##. It looks like you just have a simple algebra error buried in your work somewhere. Try writing out explicitly your result for $$\frac{1}{\alpha^2}\frac{d^2}{dx^2}\Psi$$
The offending terms should cancel in that step.
 
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I'm really dumb, I see where my mistake was thanks !
 

Related to Solving Schrödinger's Equation with a Smooth Potential Wall: A Detailed Guide

1. What is Schrödinger's equation?

Schrödinger's equation is a mathematical equation that describes the behavior of quantum particles, such as electrons, in a given system. It takes into account the wave-like nature of these particles and predicts their energy levels and probabilities of being found in different locations.

2. What is a smooth potential wall?

A smooth potential wall is a barrier or boundary in a system that has a continuous and well-defined potential energy. This means that the potential energy does not abruptly change at the wall, but rather changes gradually over a defined distance.

3. Why is it important to solve Schrödinger's equation with a smooth potential wall?

Solving Schrödinger's equation with a smooth potential wall allows us to accurately predict the behavior of quantum particles in a system. This is important in understanding and manipulating the behavior of these particles, which is crucial in fields such as quantum mechanics, materials science, and chemistry.

4. What is the process for solving Schrödinger's equation with a smooth potential wall?

The process involves using mathematical techniques, such as separation of variables and boundary conditions, to find solutions to the equation. This can be a complex process and may require numerical methods for more complicated systems.

5. Are there any limitations to solving Schrödinger's equation with a smooth potential wall?

Yes, there are limitations to this method of solving the equation. It may not be applicable to systems with non-smooth potential walls or systems with multiple particles. Additionally, it may not accurately predict the behavior of particles in highly complex systems.

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