Coefficients of a Superposition of State Vectors?

[kq6up]

Homework Statement

(f) At t = 0, a particle of mass m trapped in an infinite square well of width L is in a superposition of the first excited state and the fifth excited state, ψs(x, 0) = A (3φ1(x) − 2iφ5(x)) , where the φn(x) are correctly-normalized energy eigenstates with energies En. Which of the following values of A give a properly normalized wavefunction? $\frac { 1 }{ \sqrt { 5 } }$ $\frac { i }{ 5 }$ $\frac { i }{ \sqrt { 13 } }$ $\frac { 1 }{ 13 }$ None of these

Homework Equations

A normalized wave function: $A^{ 2 }\sum _{ i }{ a^{ * }_i a_i } =1$.

The Attempt at a Solution

Since, $\sum _{ i }{ a^{ * }_{ i }a_{ i } } =9-4=5$, $A=\frac { 1 }{ \sqrt { 5 } }$.

Does this look correct?

Thnx,
Chris

 

[blue_leaf77]

That does not look correct.
The basic definition of linear combination (or superposition) is the sum of all weighted basis states, that means if there is a negative sign in front of a particular term, that sign must belong to the expansion coefficient of that corresponding term.

 

[kq6up]

I have a negative sign there. a1*a1=9 because it is real. (-2i)*=2i, so (-2i)*x(2i)=-4 for index 5. I am not sure I follow your concern.

Chris

 

[blue_leaf77]

(-2i)*x(2i)

Shouldn't it be (-2i)*x(-2i) = 4 ?

 

[kq6up]

That would be (-2i)*x(-2i)=(2i)x(-2i)=4. Oh, ok. Simple mistake. I should have wrote it out explicitly.

It is now $A=\frac{1}{\sqrt{13}}$ which is not one of the choices. That always makes me feel insecure :D

Chris

 

[blue_leaf77]

The coefficient $A$ is in general complex, thus instead of $A^2$ you should write it $|A|^2$. Among the choices, which will give you $|A|^2 = \frac{1}{13}$?

 

[kq6up]

But it is asking for $A$, not ${|A|}^2$. Is it not?

Thanks,
Chris

 

[kq6up]

I emailed the prof who wrote this test. Yes, I see that A is in general complex, so i/sqrt(13) would satisfy ${|A|}^2=\frac{1}{13}$. Man, I am so rusty. A lot of brain sweat for what seemed to be such an easy question at first.

Thanks,
Chris

 

[blue_leaf77]

so i/sqrt(13) would satisfy ${|A|}^2=\frac{1}{13}$.

Yes that's right.

 

[kq6up]

On part g) of this question:
(g) Given the wavefunction ψs, what is the probability of measuring the energy to be E6 at t = 0? Circle one:

The correct answer would be zero, correct? That is because the coefficient in front of a state vector at level 6 would be zero. It is not included in the original wave function correct?

Thanks,
Chris

 

[blue_leaf77]

Yes the probability should be zero.

 

[kq6up]

This one should be zero too, no? The probability of finding a particle and a particular point would would be infinitesimal. You have to integrate over some range on the axis to have a finite probability.

(h) Given the wavefunction ψs, what is the probability density of finding the particle in the middle of the box at time t = 0?

0 3 9 9 Undetermined

Thanks for helping me. I have no answers to this material, and I want to make sure I have these concepts down solid.

Thanks,
Chris

 

[George Jones]

Are "probability" and "probability density" the same thing?

 

[kq6up]

Good point. Density is per unit length in a 1d scenario. Let me go back and work on it some more.

Chris

 

[kq6up]

I would be taking the derivative of $|\psi|^2$ and evaluating it at L/2, correct? I get zero because there are sine functions in both terms after finding mod squared, and taking the derivative. I am using his version of the infinite square well, where x1=0 and x2=L. See: http://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/exams/MIT8_04S13_exam1.pdf

My $|\psi|^2$ is: $|\psi |^{ 2 }=\frac { 1 }{ 13 } \left[ 9\frac { 2 }{ L } { sin }^{ 2 }(\frac { 2\pi }{ L } x)+4\frac { 2 }{ L } { sin }^{ 2 }(\frac { 6\pi }{ L } x) \right]$

Thanks,
Chris

 

[kq6up]

I would be taking the derivative of $|\psi|^2$ and evaluating it at L/2, correct? I get zero because there are sine functions in both terms after finding mod squared, and taking the derivative. I am using his version of the infinite square well, where x1=0 and x2=L. See: http://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/exams/MIT8_04S13_exam1.pdf

My $|\psi|^2$ is: $|\psi |^{ 2 }=\frac { 1 }{ 13 } \left[ 9\frac { 2 }{ L } { sin }^{ 2 }(\frac { 2\pi }{ L } x)+4\frac { 2 }{ L } { sin }^{ 2 }(\frac { 6\pi }{ L } x) \right]$

Thanks,
Chris

Never, mind. I don't know why I was thinking of taking the derivative. Mod squared is the probability density. The units are correct for that too. However, this mod square evaluated at x=L/2 should still be zero because the sine functions evaluated at multiples of $\pi$ are equal to zero.

Correct?

Thanks,
Chris