Simple Quantum Mechanics Question - Express Wave Function

[samoth1]

As is always my problem with physics homework, I am probably thinking to hard about this... however, I am not sure how to express this wave function!

This is the question:

24) The time independent wave function of a particle is given in the graph below. The function rises linearly from the origin with slope +k.

http://uwoshkosh.elitefitness.us/Mod%20Phys1.JPG

This is a single sawtooth. We covered how to express a continuous square wave, but I don't know how to get this wave function.

The two questions accompanying this are to find the probability in a specified range, and to find the expectation value of the particle's kinetic energy. These two I know how to do... it's just the wave function that has me baffled!


Thanks for any help!

 

[cookiemonster]

Isn't that a graph of the psi function...? Just square it and integrate to get the probability distribution.

cookiemonster

 

[Doc Al]

As is always my problem with physics homework, I am probably thinking to hard about this... however, I am not sure how to express this wave function!

It's a straight line between 0, a. Zero elsewhere. What's the equation for a straight line?

Don't forget that the wavefunction must be normalized.

 

[samoth1]

Okay... I think I have an idea of what's going on now. I was thinking I needed a function representing a sawtooth wave, or something to that effect.

Regarding this:

It's a straight line between 0, a. Zero elsewhere. What's the equation for a straight line?

We did an example similar to this in lecture, however it was a square wave pulse of width L -- a straight line from 0 to L. This new example here had/has me a tad confused as how to represent the wave function.

So, I have that this wave function should be something like

psi(x)=((2 pi)/L) x

However... another example (classical) showed something similar to this, but of the form

psi(x)=sin ((2 pi)/L) x

I am unsure about this part... we have had so many new and different things thrown at us in the last few weeks, I find myself confused as to what's what anymore!



As a sidenote: This is part of a Modern Physics course, so we have only been spending about 2-3 weeks on quantum mechanics. This is my first real exposure to QM, and a hurried one at that, so I apologize for lacking many basic aspects.

 

[samoth1]

Hmm... what I'm trying to say, really, is that I am uncertain how to represent the wave function that needs to be normalized.

I am confident in the processes... but being that I am new to the subject, given a problem different than that shown in lecture raises a lack of confidence in the initial step representing the problem (especially given that the initial representation lays the foundation for the remaining 95% of the problem).


Thank you for the replies thus far; they have indeed helped in my understanding. Still... I don't feel confident in my approach to the wave function.

 

[Doc Al]

So, I have that this wave function should be something like

psi(x)=((2 pi)/L) x

Here's how I would do it. It's just a straight line going through the origin, so (for x between 0 and a):
$$\psi(x) = kx$$
Normalization requires:
$$\int_{0}^{a} \psi^*(x) \psi(x)dx = 1$$
So, use this to figure out what k must be. Make sense?

 

[samoth1]

That is what I did, however I used

psi(x)=((2 pi)/L) x

instead of

psi(x)=kx

so that the function explicitly showed respect to L and x. I then normalized from there. Is that acceptable?



We spent so much time talking about different waves in both classical and quantum mechanics (all on different axes!) that for some reason, I thought there was a special function needed for this to show a sawtooth wave function, as the professor discussed sawtooth, square, and other waves. This makes sense, though.

Thank you for clarifying!

 

[Doc Al]

That is what I did, however I used

psi(x)=((2 pi)/L) x

What is L? Where does the pi come from?

When all is said and done, there is only one acceptable answer for psi(x). It must be (some constant)X. Find that constant!

 

[samoth1]

What is L? Where does the pi come from?

I used k=(2 pi)/ L

Where L is the width (expressed as 'a' in the problem, but I see L used much more)... another part of the question asks to find the probability of finding a particle in the range of x=(1/4)a to x=(3/4)a.

Am I using/describing L ('a') correctly? Nearly every problem we discuss involves length L, whether in the context of the problem here, or in a square well with walls at, for example, 0 and L, or -L/2 and +L/2.

 

[Doc Al]

I used k=(2 pi)/ L

Where L is the width (expressed as 'a' in the problem, but I see L used much more)... another part of the question asks to find the probability of finding a particle in the range of x=(1/4)a to x=(3/4)a.

Am I using/describing L ('a') correctly?

When they describe the wave function using "L", use "L". In this case they used "a", so you'd better use "a". Go back to post # 6 and do what I suggested. Solve for k.

Hint: k ≠ (2 pi)/ L or (2 pi)/ a