What Are the Coefficients for Probability in Quantum Mechanics?

[jplcarpio]

Hi all,


In my class we were given an assignment from Introduction to Quantum Mechanics by David Griffiths. The question is in Chapter 3, problem 3.34.

[PLAIN]http://img41.imageshack.us/img41/7899/prob334.png

(The system has a harmonic oscillator potential.)

Right now I'm stuck with getting the probability for the measured energy values. I know that the probability is the square of the absolute value of the coefficients of the energy eigenstates. Based on the question, that would be:

[PLAIN]http://img59.imageshack.us/img59/3004/prob3341.png

And since the measurements are obtained with equal probability:

[PLAIN]http://img14.imageshack.us/img14/3919/prob3342.png



My question is, what are the coefficients? I know it would be 1 over the square root of 2, I'm just not sure of the sign. Wouldn't the square of the absolute value imply that there are positive and negative values?

[PLAIN]http://img508.imageshack.us/img508/7591/prob3343.png

Thanks so much!

 

[Fredrik]

If c is a complex number and f a wavefunction, cf is a wavefunction that represents the same state as f. (If we require all wavefunctions to be normalized, the above only holds when |c|=1). So if you can take your coefficients to be $1/\sqrt{2}$, you can also take them to be (for example) $-i/\sqrt{2}$. There's an absolute value function in the formula you're using because it's needed in general. It's not needed in those special cases where all the coefficients are real and positive.

 

[jplcarpio]

If c is a complex number and f a wavefunction, cf is a wavefunction that represents the same state as f. (If we require all wavefunctions to be normalized, the above only holds when |c|=1). So if you can take your coefficients to be $1/\sqrt{2}$, you can also take them to be (for example) $-i/\sqrt{2}$. There's an absolute value function in the formula you're using because it's needed in general. It's not needed in those special cases where all the coefficients are real and positive.

It wasn't specified that the coefficients are real and positive for the problem, though in the discussions the book Griffiths assumed that they were for simplicity.

How would you know if $1/\sqrt{2}$ or $-i/\sqrt{2}$ should be used (based on your example), if c is indeed complex?

Thank you again. :)

 

[Fredrik]

It doesn't matter which one you use, because if $f=\frac{1}{\sqrt{2}}(u_1+u_2)$ and $g=\frac{-i}{\sqrt{2}}(u_1+u_2)$, then f and g represent the same state (since g=-if), and they both give you the same final result. (If they didn't, they couldn't possibly represent the same state).

 

[vela]

I haven't actually worked out the problem, but I think the point of the problem is to find the relative phase of the constants. The best you can do at this point is to arbitrarily set one to $1/\sqrt{2}$. You can do this for the reason Fredrik has explained. For the other, you'll have to include a complex phase factor, so it'll be $e^{i\phi}/\sqrt{2}$. Presumably <p> will vary as a function of Φ, which will allow you to answer the remaining questions in the problem.

 

[mathfeel]

My question is, what are the coefficients? I know it would be 1 over the square root of 2, I'm just not sure of the sign. Wouldn't the square of the absolute value imply that there are positive and negative values?
Thanks so much!

You know far less than that. All you know is that the magnitude of both coefficients are 1/sqrt(2), but you do not know the relative phase. So make that your unknown variable to find the maximal <p> against.

 

[jplcarpio]

Ohhh, okay. I didn't quite understand how to incorporate the complex phase factor into the coefficients, we weren't given examples of that.

Thank you all, I believe I can finish the problem now. :)