Particle in one dimensional potentional well

[derp267]

I hope this is in the right place, I'm new here. Anyway, my teacher hasn't shown us an example where U is anything but infinity, Uo, or 0 and I'm completely stumped on part B for this question since U is a function of x

Homework Statement

A particle of mass m moves in a one-dimensional potential well:
U(x)={infinity...x<0
{-hbar^2/mbx...x>=0

The normalized wave function is:
Ψ (x)={0...x<0
{Axe^(-x/b)...x>=0

Where b and A are constants.
a) Describe in words or equations how you would evaluate A( you do not need to actually evaluate for A).
b) Prove that the above Ψ (x) for x>=0 is an acceptable wave function
c) Find the total energy of the particle. Express your answer in the simplest terms

Homework Equations

The Attempt at a Solution

I did this on paper already and I don't know how to type an integral or anything so I just made a picture..apologies for the bad handwriting
http://imgur.com/OV8RN

So do I make -2mE/hbar^2 -k^2? Then I'm still left with -2/bx and I don't think I can use eulers method with that..I'm stuck

 

[ideasrule]

Assuming you didn't make any algebra errors, your math is correct so far. Since you're trying to prove that Ψ (x) is a solution for x>0, plug their wavefunction into your final equation and see if the left and right sides equal one another. Remember that you can adjust the constant E to try to make the two sides equal, but E is a constant, so it can't depend on x.

 

[derp267]

Thanks for your relpy. Okay so I did as you said and plugged in Axe^(-x/b) for Ψ and did some algebra. Then I did the second derivative of Ψ(x) and set them equal to each other.

What I end up with is the following:
http://i.imgur.com/mflUG.png

So does that mean that E=(-hbar^2)/(2mb^2)?

 

[ideasrule]

Assuming your algebra is right, yes. That's the energy which corresponds to the given wavefunction.

 

[paranormal]

I would like to first commend you for seeking help and clarification on this topic. It is important to fully understand the concepts and equations involved in order to successfully solve problems and conduct research in the field of physics.

Regarding the particle in a one-dimensional potential well, it is important to understand the concept of a potential well and how it affects the motion and energy of a particle. In this case, we have a potential well that is infinite for x<0 and decreases as x increases for x>=0. This means that the particle is confined to the region x>=0 and cannot exist for x<0.

To evaluate the constant A in the normalized wave function, we can use the normalization condition, which states that the integral of the square of the wave function over all space must equal 1. In this case, since the wave function is 0 for x<0, we only need to integrate over the region x>=0. Using the given wave function, we can set up the integral as follows:

1 = ∫|Ψ(x)|^2 dx
1 = ∫A^2x^2 e^(-2x/b) dx
1 = A^2 ∫x^2 e^(-2x/b) dx

At this point, we can use integration by parts to evaluate the integral. After solving for A, we can plug in the value of A into the wave function to get the normalized wave function.

To prove that the above Ψ (x) for x>=0 is an acceptable wave function, we can use the Schrödinger equation, which describes the time evolution of a quantum system. The Schrödinger equation for this system is given by:

-ĤΨ(x) = EΨ(x)

Where Ĥ is the Hamiltonian operator, E is the energy of the particle, and Ψ(x) is the wave function. By plugging in the given wave function, we can show that it satisfies the Schrödinger equation for x>=0. This proves that it is an acceptable wave function for this system.

Finally, to find the total energy of the particle, we can use the energy operator, which is given by:

Ê = -ĥ^2/2m * d^2/dx^2 + U(x)

By plugging in the given potential function, we can solve for the energy of the particle