# Solve 3D TISE in Potential Well: Eigenvals/Fns & Normalize

• obnoxiousris
Remember to use the full expression for the wavefunction, not just the trigonometric part. In summary, when solving the TISE for a particle in a 3D cubical potential well, both the trigonometric and complex solutions can be used. The full expression for the wavefunction should be used, including both eikx and e-ikx.
obnoxiousris

## Homework Statement

A particle in a 3D cubical potential well. The walls are Lx, Ly, Lz long.
Inside the well, V(x,y,z)=0 when 0<x<Lx, 0<y<Ly, 0<z<Lz. V= ∞ elsewhere.
Solve the TISE to find the eigenfunctions and eigenvalues of this potential. And to normalise the wavefunctions.

(Hint: look for separable solutions in the form ψ(x,y,z)=X(x)Y(y)Z(z))

## Homework Equations

(-hbar/2m)(∇2)ψ(x,y,z)+V(x,y,z)ψ(x,y,z)=Eψ(x,y,z)

## The Attempt at a Solution

I'm confused about which equation to use for this question. I know that for a potential well, the V terms are zero, and i can write -hbar/2m and E in terms of k, so the equation reduces to ∇2ψ(x,y,z)=-k2ψ(x,y,z).

we did the one dimensional well in class. my teacher used ψ(x)=Asin(kx)+Bsin(kx) as a general solution, then subbed in the boundary conditions like when x=L, ψ(x)=0 and so on.

However, I know that ψ(x)=Aeikx is also a general solution to the differential equation. And this is used for free particles. I did some research online, and apparently I can treat a particle in the well as a free particle.

So i don't know which equation i should use, and what are the differences between the two equations? Some sites I've seen used the trig, some used the complex.

I tried to use the complex one, I got (after normalisation) ψ(x,y,z)=(1/sqrt3L)ei(kxx+kyy+kzz). Then I tried to sub in the boundary conditions, only to realize no matter what i did i couldn't make ψ=0.
So I tried to use the trig, but the equation wasn't separable.

I really don't know what to do here, some help would be appreciated! Thanks!

Last edited:
There are two independent solutions, eikx and e-ikx. You will need both in order to satisfy your boundary conditions.

## What is the Time-Independent Schrödinger Equation (TISE) in 3D?

The Time-Independent Schrödinger Equation (TISE) is a mathematical equation used in quantum mechanics to describe the behavior of a quantum system in a stationary state. It is a partial differential equation that relates the wave function of a system to its energy. In 3D, it takes into account the three spatial dimensions.

## What is a Potential Well in quantum mechanics?

A Potential Well is a region in space where the potential energy of a particle is lower than the energy outside of the region. In quantum mechanics, a particle trapped in a potential well has a higher probability of being found within the well than outside of it. This is used to model various physical systems, such as atoms and molecules.

## How do you solve the TISE in a Potential Well?

To solve the TISE in a Potential Well, you must first set up the TISE in its general form, which includes the potential energy function for the specific well. Then, you can use various mathematical techniques, such as separation of variables or the shooting method, to solve the TISE and obtain the eigenvalues (or energy levels) and eigenfunctions (or wave functions) for the system.

## What are eigenvalues and eigenfunctions in the context of the TISE in a Potential Well?

Eigenvalues are the energy levels that a particle can possess in a Potential Well, while eigenfunctions are the corresponding wave functions that describe the spatial distribution of the particle in the well. These quantities are obtained by solving the TISE and are used to characterize the behavior of the quantum system.

## How do you normalize the eigenfunctions obtained from solving the TISE in a Potential Well?

To normalize the eigenfunctions, you must use the normalization condition, which states that the integral of the squared magnitude of the wave function over all space must equal 1. This ensures that the wave function represents a valid probability distribution. To normalize the eigenfunctions, you may need to use techniques such as the Gram-Schmidt process or apply boundary conditions.

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