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

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• paul159753
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.
paul159753
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 !

Last edited:
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.

paul159753
I'm really dumb, I see where my mistake was thanks !

## 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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