Understanding the Quantization of Azimuthal Wavefunctions in Quantum Mechanics

In summary: So in summary, the azimuthal part of the wavefunction \Psi(\phi) = Ae^{-iq\phi} must have a value of q that is an integer in order for the wavefunction to be normalized. This can be shown by setting up and solving the integral \int \Psi^* \Psi d\phi = 1 and finding that q must satisfy both \cos (-q2 \pi) = 1 and \sin(-q2 \pi) = 0. This ultimately leads to the conclusion that q must be an integer. Additionally, the value of A can be found to be \sqrt{\frac{1}{2\pi}} by solving the normalization integral. The value of L_z for this particle is
  • #1
Nylex
552
2
I have a question that I'm struggling with a bit.

The azimuthal part of the wavefunction of a particle is

[tex]\Psi(\phi) = Ae^{-iq\phi}[/tex] where [tex]\phi[/tex] is the azimuthal angle. Show that q must be an integer. By normalising the wavefunction, find the value of A. What is the value of L_z for this particle?

Ok, I know that [tex]\Psi(\phi) = \Psi(\phi + 2\pi)[/tex] because [tex]\phi[/tex] and [tex]\phi + 2\pi[/tex] are the same angle.

So, [tex]Ae^{-iq\phi} = Ae^{-iq(\phi + 2\pi)}[/tex]

and [tex]Ae^{-iq\phi} = Ae^{-iq\phi}e^{-iq2\pi}[/tex]

[tex]\Rightarrow e^{-iq2\pi} = 1[/tex]

How does this imply that q is an integer? This was the way it was done in lectures, but we were just told that this shows q is an integer. I thought it was something to do with [tex]e^{ix} = \cos x + i\sin x[/tex], but I'm not sure.

For the normalising bit, I know I need to use [tex]\int \Psi^* \Psi d\phi = 1[/tex] but I'm not sure about the limits. This is what I've done:

[tex]\int \Psi^* \Psi d\phi = 1[/tex]

[tex]\int_{0}^{2\pi} Ae^{iq\phi}Ae^{-iq\phi} = 1[/tex]

[tex]A^2 \int_{0}^{2\pi} d\phi = 1[/tex]

So [tex]A = \sqrt{ \frac{1}{2\pi} }[/tex]

Is this correct? As for the angular momentum component, I'm working on it.

Thanks.
 
Physics news on Phys.org
  • #2
Sure,that's the Condon-Shortley convention.Actually the wave function is a phase factor;so other one would be superfluous.

[tex] e^{-iq2\pi}=\cos\left(-q2\pi\right)+i\sin\left(-q2\pi\right)=1 [/tex]

So when is the cosine =1 ...?(Don't worry,the sine in those points is automatically 0)

Daniel.
 
  • #3
That's where I was getting confused. How do you know sine is 0 there? I know [tex]\sin n\pi = 0[/tex] where n is an integer, but if you don't know n is an integer in the first place, how can you assume that those sine terms are 0?
 
  • #4
If the cosine is "+1" (as it should be),then automatically the sine is 0,because we know that

[tex] \sin^2 x+\cos^2 x=1 \, \ x\in\mathbb{R} [/tex]

Daniel.

P.S.As i said,don't worry about the sine.
 
  • #5
Here's another way to look at it: [itex]q[/itex] must satisfy both of the following conditions:

[tex]\cos (-q2 \pi) = 1[/tex]

[tex]\sin(-q2 \pi) = 0[/tex]

If we start with the first condition, that eliminates all values of [itex]q[/itex] except the + and - integers, and zero. These remaining values of [itex]q[/itex] all satisfy the second condition, so we're done.

Alternatively, we can start with the second condition. In this case, we eliminate all values of [itex]q[/itex] except the + and - integers and half-integers, and zero. Now we apply the first condition to those remaining values, which eliminates the half-integers, and gives us the same final result as before.
 
  • #6
Ahh ok, thanks.
 

Related to Understanding the Quantization of Azimuthal Wavefunctions in Quantum Mechanics

1. What is quantum mechanics and why is it important?

Quantum mechanics is a branch of physics that studies the behavior of particles at a very small scale, such as atoms and subatomic particles. It is important because it helps us understand the fundamental laws of nature and has practical applications in fields such as technology, medicine, and energy production.

2. What is the Schrödinger equation and how does it relate to quantum mechanics?

The Schrödinger equation is a mathematical equation that describes how the quantum state of a physical system changes over time. It is a fundamental equation of quantum mechanics and helps us understand the behavior of particles at a microscopic level.

3. Why is it difficult to visualize or understand quantum mechanics?

Quantum mechanics operates on a scale that is vastly different from our everyday experience, making it difficult to visualize or understand. It also challenges our classical notions of cause and effect, as particles can exist in multiple states at the same time.

4. What is the uncertainty principle and how does it relate to quantum mechanics?

The uncertainty principle states that it is impossible to know the exact position and momentum of a particle simultaneously. This principle is a fundamental aspect of quantum mechanics and demonstrates the probabilistic nature of particles at a subatomic level.

5. What are some real-world applications of quantum mechanics?

Quantum mechanics has many real-world applications, such as in the development of transistors for electronics, cryptography for secure communication, and medical imaging techniques like MRI. It also has potential applications in quantum computing, which could greatly impact fields such as drug discovery and optimization problems.

Similar threads

  • Introductory Physics Homework Help
Replies
6
Views
317
  • Introductory Physics Homework Help
Replies
4
Views
809
  • Introductory Physics Homework Help
Replies
17
Views
455
  • Introductory Physics Homework Help
Replies
6
Views
961
  • Introductory Physics Homework Help
Replies
4
Views
1K
  • Introductory Physics Homework Help
Replies
15
Views
1K
  • Introductory Physics Homework Help
Replies
6
Views
1K
  • Introductory Physics Homework Help
Replies
3
Views
269
  • Introductory Physics Homework Help
Replies
28
Views
464
  • Introductory Physics Homework Help
Replies
2
Views
1K
Back
Top