Probability Density for Wavefunctions undergoing phase shifts.

In summary, the problem is that the student is trying to solve for the trig identities using Euler's formula, but is having difficulty because they are missing one of the trig identities. They need to use the cos²(x) + sin²(x) = 1 trig identity to solve for the other two.
  • #1
Einstein2nd
25
0
Sorry for not using template but you should find everything in the image provided:

Hey guys. All of the info for the problem is in a picture.

I've tried working on this for ours and I still can't seem to get the trig identities right :(

http://img208.imageshack.us/img208/1770/assignmentquestion2.jpg

NOTE THAT THERE SHOULD BE ANOTHER BRACKET ON THE VERY END OF THE EQUATION FOR THE PROBABILITY DENSITY. IT SHOULD HAVE sin(delta)), NOT sin(delta) AS IT CURRENTLY HAS.

from that final step, I've done many things by both hand and scientific notebook and I just can't seem to get things to simplify down properly. There is no way I could possibly post all of the different things I've tried but don't worry, I'm not simply looking for a copy-paste answer into homework. I want to be able to understand the working.

Please clarify my initial working and steer me in the correct direction. I'm pretty sure that I understand the physics, it's just the maths...
 
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  • #2
Hi Einstein2nd,

What you've done so far is correct. Now use Euler's formula exp(ix) = cos(x) + i sin(x). Once you've separated the real and imaginary parts using this formula, add their squares to get the answer.

The only trig identity you will need is cos²(x) + sin²(x) = 1.
 
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  • #3
Thank you for your help so far. I'm not getting the right answer though. When doing your above procedure. I'm using Euler's formula on what I have inside the brackets in the last line of my opening post. This gives me (1 + icos(delta) - sin(delta)) because there was that existing i out the front.

I separated the real and imaginary parts and squared each of them (not sure why this is legit) to get:

(1-sin(delta))^2 + (icos(delta))^2 = -cos2(delta) + sin2(delta) - 2sin(delta) + 1

this either isn't correct or I'm missing another step of simplification. I can't use that Pythag identity yet.
 
  • #4
Hi,

It should be (-1 + i cos(δ) - sin(δ)), you forgot the minus before 1. And the absolute value squared of a complex number (x + iy) is x² + y², not x² + (iy)².
 
  • #5
so you're saying I should do:

|(-1-sin(delta))^2| + |(icos(delta)^2|

That does give me sin2(delta) + 2sin(delta) + 1 + cos(delta) if I do it as you say.

What I'd actually found before your most recent post is if I multiply what I have there by its complex conjugate. Please check if this is right:

(-1+ie^(i delta)) * (-1 - ie^(-i delta)

=ie^(-iδ) -ie^(iδ) + e^(-iδ)e^(iδ)+1

=2sinδ+(cosδ-isinδ)(cosδ+isinδ)+ 1

=2sinδ+2

That gives me 2sin(delta) + 2

I also tried multiplying with complex conjugate after taking into sin and cos first.

(icosδ - sinδ - 1)(icosδ - sinδ - 1)

=cos²δ + sin²δ + 2sinδ + 1

=2sinδ+2

Is what I've done alright? All I did was put a negative in front of i wherever it came up (twice when in exponential form, once when already converted to trig first)
 
  • #6
Yes it looks correct.
 
  • #7
Thank you very much for your time.
 

1. What is probability density for wavefunctions undergoing phase shifts?

The probability density for wavefunctions undergoing phase shifts is a measure of the likelihood of finding a particle at a specific location in space. It is described by the square of the wavefunction, which represents the amplitude of the particle's wave at that location.

2. How do phase shifts affect the probability density of a wavefunction?

Phase shifts can cause the probability density of a wavefunction to change in space. This occurs when the phase of the wavefunction changes, leading to constructive or destructive interference at different points in space.

3. What is the mathematical formula for calculating probability density for wavefunctions undergoing phase shifts?

The mathematical formula for calculating probability density is given by the absolute square of the wavefunction, which is represented as |Ψ|². This takes into account both the magnitude and phase of the wavefunction.

4. How does the probability density of a wavefunction change when undergoing multiple phase shifts?

When a wavefunction undergoes multiple phase shifts, the probability density can change significantly. This is because each phase shift can cause constructive or destructive interference, leading to peaks and troughs in the probability density at different points in space.

5. What is the significance of probability density for wavefunctions undergoing phase shifts in quantum mechanics?

The probability density for wavefunctions undergoing phase shifts is a fundamental concept in quantum mechanics. It allows us to calculate the probability of finding a particle at a specific location in space, which is crucial for understanding the behavior of quantum systems and making predictions about their future states.

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