# Non trivial solution to Schrödinger equation for 1-D infinite well

• MrMuscle
In summary: As for modifying the code, you can try solving for one of the boundary conditions instead of using both.
MrMuscle
TL;DR Summary
Hello, I am trying to find the solution of Schrödinger equation on matlab. However, when I apply boundary conditions, MATLAB only gives me the solution with both coefficients 0. I want to find the solution : Asin(n*pi*x/L)
You can see my code below. Could you please tell me where is my mistake?
Thanks in advance!
Hello, I am trying to find the solution of Schrödinger equation on matlab. However, when I apply boundary conditions, MATLAB only gives me the solution with both coefficients 0. I want to find the solution : Asin(n*pi*x/L)
You can see my code below. Could you please tell me where is my mistake?
Thanks in advance!

clear,clc %Solution of Schrödinger eqn with 1-D infinite well
syms WF(x) V(x) %WF is the wavefunction, V(x) is the potential
syms hbar m L E positive %E energy, L length of the well, m mass, hbar reduced Planck
V(x)=0 %Potential
Sch=((-hbar^2)/(2*m))*diff(WF,x,2)+V*WF==E*WF %Schrödinger equation

Solution1=dsolve(Sch) %Solution without applying boundary conditions

%BOUNDARY CONDITIONS
cond1=WF(0)==0;
cond2=WF(L)==0;
conds=[cond1 cond2];
Solution2=dsolve(Sch,conds) %Solution with applying boundary conditions.

Matlab is trying to solve your setup for an arbitrary energy ##E##. This will generally not have a solution but ##E## will be quantised. What you want to do is to solve the differential equation using one of the boundary conditions, leaving you one integration constant. Then you can solve the second boundary condition for the energy. This will give you the quantised energy levels.

Note that MATLAB is not the ideal for symbolic computations.

MrMuscle and PeroK
Thanks for the answer! what would you suggest for symbolic calculations?
Also, do you have any suggestions about how to modify the code?

Personally, I would use Mathematica. Maple is also popular.

## What is the Schrödinger equation for a 1-D infinite well?

The Schrödinger equation for a 1-D infinite well describes the behavior of a particle confined to a one-dimensional region with infinite potential barriers on either side. It is given by:

where Ĥ is the Hamiltonian operator, ψ(x) is the wave function of the particle, E is the energy of the particle, and x is the position along the well.

## What is the significance of a non-trivial solution to the Schrödinger equation for a 1-D infinite well?

A non-trivial solution to the Schrödinger equation for a 1-D infinite well means that the wave function ψ(x) is not equal to zero everywhere along the well. This indicates that there is a non-zero probability of finding the particle in certain regions of the well, which is necessary for the particle to have physical significance.

## How is the energy of the particle related to the non-trivial solutions of the Schrödinger equation for a 1-D infinite well?

The energy of the particle is directly related to the non-trivial solutions of the Schrödinger equation for a 1-D infinite well. The allowed energy levels of the particle are determined by the solutions to the equation, with each solution corresponding to a different energy state. The lowest energy state corresponds to the ground state of the particle.

## What is the physical interpretation of the wave function in the Schrödinger equation for a 1-D infinite well?

The wave function ψ(x) in the Schrödinger equation for a 1-D infinite well represents the probability amplitude of finding the particle at a particular position x along the well. The square of the wave function, |ψ(x)|², gives the probability density of finding the particle at that position. This is known as the Born interpretation of the wave function.

## How do the potential barriers affect the solutions of the Schrödinger equation for a 1-D infinite well?

The potential barriers in the 1-D infinite well affect the solutions of the Schrödinger equation by restricting the allowed energy levels of the particle. The particle can only exist within the well, so the solutions must satisfy the boundary conditions at the potential barriers. This leads to the quantization of energy levels in the well, with only certain energy values being allowed for the particle.

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