# Wave function in Infinite/Finite Potential Wells

• lee_sarah76
In summary, ψ=A*sin(kx) + B*cos(kx) is the functional form of the wave function in the ground state in the five regions x<0, 0<x<a, a<x<b, b<x<L, and x>L.
lee_sarah76

## Homework Statement

What is the functional form of the wave function in the ground state in the
five regions x<0, 0<x<a, a<x<b, b<x<L, and x>L?

I've attached the picture of the potential well as well here:

## Homework Equations

Schrodinger time independent equation

## The Attempt at a Solution

I understand that for x < 0 and x > L, ψ = 0 because there's no possibility of the particle being there.

And then between a and b: ψ = A*sin(kx) + B *cos(kx) where k = √(2mE)/(hbar)2

It's between 0 < x < a and b < x < L that I'm having issues. Can anyone point me in the right direction?

Thanks!

So remember how you get k in the first place in the region where the potential is 0:

$\frac{-\hbar^2}{2m}\frac{∂^2\psi}{∂x^2}=E\psi$

The solution comes from 'guessing' $ψ = A\sin(kx) + B\cos(kx)$, and plugging this into the differential equation and algebraically solving for k.

Now consider the regions where $V=U_0$

$\frac{-\hbar^2}{2m}\frac{∂^2\psi}{∂x^2}+U_0\psi=E\psi$

$\frac{-\hbar^2}{2m}\frac{∂^2\psi}{∂x^2}=(E-U_0)\psi$

Notice how this is the same equation as above, the only difference being $E-U_0$ instead of $E$.

At this point you just assume a similar solution for $\psi$ in the two regions with potential $U_0$ as you did with in the 0 potential region. Note however that the coefficients infront of the sines and cosines in each of the 3 regions are different in each region; to solve for each one continuity conditions, boundry values and normalizationmust be applied.

You should consider if the the ground state wavefunction will be sinusoidal or exponential in regions 0<x<a and b<x<L.

So I get that it will be the same as the function for a < x < b except that the constants in front of the sine/cosine will be different and k = √2m(E - Uo)/ (hbar)2?

Is that right?

Maybe, maybe not. Did you consider what TSny said? How did you conclude the wave function would sinusoidal and not exponential in that region?

The others have a point; the solutions you 'should' assume had the form

$\psi(x)=Ce^{zx}+De^{-zx}$

For some z that may end up being real, imaginary or complex; this is how you get the original ('particle in a box') solution in the first place.

Okay I think I see that. But how do we determine if the wavefunction will be sinusoidal or exponential?

lee_sarah76 said:
Okay I think I see that. But how do we determine if the wavefunction will be sinusoidal or exponential?

Recall the finite square well problem. What is the condition to get exponential behavior of the wavefunction outside the well?

Oh I see. So if the particle is bound by the potential, we know V > E, so we "guess" the exponential solution. But if the particle is free and E > V, then we guess the sinusoidal solution? Is that the correct interpretation?

Yes, although I wouldn't say it involves "guessing". If you study the form of the Schrodinger equation for the case where Vo > E, then you can see that the solution must be exponential. (But maybe you're using "guessing" more loosely.) Anyway, you're on the right track!

Ah, yes, I'm using "guessing" in the the sense that as I'm learning differential equations, the book tells us to "guess" solutions to the second order differential equations. It's not really guessing, but I've gotten used to the term.

Thanks so much!

## 1. What is a potential well?

A potential well is a region in space where the potential energy of a particle is lower than its surrounding regions. This can be visualized as a "well" where the particle is confined to a certain area due to the potential energy barrier surrounding it.

## 2. What is the difference between an infinite and finite potential well?

An infinite potential well is a theoretical concept where the potential energy barrier surrounding the particle is infinitely high, meaning the particle is completely confined to a specific region. A finite potential well, on the other hand, has a potential energy barrier that is not infinitely high, allowing the particle to have a non-zero probability of existing outside of the well.

## 3. What is the wave function in a potential well?

The wave function in a potential well describes the behavior and probability of a particle existing in the well. It is a mathematical function that represents the particle's position and momentum at any given time. The shape of the wave function is dependent on the potential energy barrier of the well.

## 4. How does the particle's energy relate to the potential well?

The particle's energy is determined by the potential energy barrier of the well. In an infinite potential well, the particle's energy is quantized, meaning it can only have certain discrete values. In a finite potential well, the particle's energy can have a continuous range of values.

## 5. What is the significance of the wave function's boundary conditions in a potential well?

The boundary conditions of the wave function in a potential well determine the allowed energy levels and the shape of the wave function. In an infinite potential well, the boundary conditions require the wave function to be zero at the boundaries, resulting in the quantized energy levels. In a finite potential well, the boundary conditions allow for the wave function to have a non-zero value at the boundaries, resulting in a continuous range of energy levels.

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