Quantum mechanics question.

In summary, the problem at hand involves showing that the classical velocity of a particle can be expressed as v = d<x>/dt = 1/m<p-qA>, and the expectation value of the time derivative of x is given by <v> = ∫Ψ* (x d/dt) Ψ. To solve this, one must use the commutator [H,x] and the substitution p = (hbar/i)∇ - qA, where p is momentum, q is charge, and A is the vector potential. This leads to the Hamiltonian being written as [1/2m (p-qA)^2 + qϕ]. The solution also involves taking the commutator of the Hamilton
  • #1
curious george
11
1
I was recently given a problem that asked me to show that the classical velocity of a particle is given by:

[tex] v = \frac{d \langle x\rangle}{dt} = \frac{1}{m} \langle({\bf p}-q{\bf A})\rangle[/tex]

The expectation value of the time derivative of x is given by:

[tex]\langle v\rangle = \int{\Psi ^{*} (x \frac{d}{dt}) \Psi[/tex]

So then I just work this out, and what do I do after that? How do I get from here to the form the problem is asking for?
 
Last edited:
Physics news on Phys.org
  • #2
I assume p and q are the "generalized coordinates" (p is "x-position" and q is momentum) but what is A?

Also you are using "v" to mean two different things. In the first equation, v is the "classical" velocity while in the second it is the velocity "operator".
In other words, the v in the first equation IS the <v> of the second.
 
  • #3
HallsofIvy said:
I assume p and q are the "generalized coordinates" (p is "x-position" and q is momentum) but what is A?

Also you are using "v" to mean two different things. In the first equation, v is the "classical" velocity while in the second it is the velocity "operator".
In other words, the v in the first equation IS the <v> of the second.

I'm sorry, this material is new to me and I'm not sure what is "standard" notation and what isn't. The question was asked in the context of gauge invariance in electrodynamics(that's what we're discussing in my class right now), so I believe the whole expression "p-qA" is the substitution:

[tex]\bf{p} = \frac{\hbar}{i} \bigtriangledown \to \bf{p} - q\bf{A}[/tex]

so p is momentum, q is charge, and A is the vector potential. We also have the substitution:

[tex]i\hbar\frac{\partial}{\partial t} \rightarrow i\hbar\frac{\partial}{\partial t} - q\phi[/tex]

Where q is again the charge and phi is the scaler potential.

So the schrodinger equation should look like:

[tex]i\hbar\frac{\partial\Psi}{\partial t}= \left[\frac{1}{2m} \left(\frac{\hbar}{i}\bigtriangledown + qA\right)^2+q\phi\right]\Psi[/tex]
 
Last edited:
  • #4
Start with the commutator [tex] [H,x] [/tex], from there you should be able to work out the solution.

dt
 
  • #5
Dr Transport said:
Start with the commutator [tex] [H,x] [/tex], from there you should be able to work out the solution.

dt


So the hamiltonian in this case is
[tex]\left[\frac{1}{2m} \left(\frac{\hbar}{i}\bigtriangledown + qA\right)^2+q\phi\right][/tex]
right?

So if I take the commutator of that with x:
[tex]\left[\left[\frac{1}{2m} \left(\frac{\hbar}{i}\bigtriangledown + qA\right)^2+q\phi\right],x\right][/tex]

I should be able to figure out the time derivative of <x> by

[tex]\frac{d\langle x \rangle}{dt}=\frac{i}{\hbar}\left\langle\left[\left[\frac{1}{2m} \left(\frac{\hbar}{i}\bigtriangledown + qA\right)^2+q\phi\right],x\right]\right\rangle+\frac{\partial \langle x \rangle}{\partial t}[/tex]

Right? I just want to make sure I've got the idea right before I try to go and do the math. Thanks for the help guys.
 

1. What is quantum mechanics?

Quantum mechanics is a branch of physics that studies the behavior of matter and energy at the atomic and subatomic level. It describes how particles behave and interact with each other, and is the foundation of modern physics.

2. How does quantum mechanics differ from classical mechanics?

Classical mechanics is the branch of physics that deals with the motion of large objects, while quantum mechanics deals with the behavior of particles at the atomic and subatomic level. In classical mechanics, objects have definite positions and velocities, while in quantum mechanics, particles can exist in multiple states at the same time and have uncertain positions and momenta.

3. What is the uncertainty principle in quantum mechanics?

The uncertainty principle is a fundamental principle in quantum mechanics that states that it is impossible to measure certain pairs of physical properties (such as position and momentum) of a particle simultaneously with perfect accuracy. This is because the act of measuring one property affects the other property, making it impossible to know both values with complete certainty.

4. What are quantum states and how do they relate to quantum mechanics?

Quantum states are a mathematical description of the properties and behavior of a quantum system. In quantum mechanics, particles can exist in multiple states at the same time, and the mathematical formalism of quantum states allows us to accurately describe and predict the behavior of these particles.

5. How is quantum mechanics applied in real-world technologies?

Quantum mechanics has many practical applications in modern technology, including transistors, lasers, and computers. It is also the basis for technologies such as MRI machines, solar cells, and quantum cryptography. Additionally, ongoing research in quantum computing and quantum communication has the potential to revolutionize computing and communication technologies in the future.

Similar threads

  • Introductory Physics Homework Help
Replies
2
Views
307
  • Quantum Physics
Replies
13
Views
949
  • Introductory Physics Homework Help
Replies
6
Views
231
  • Introductory Physics Homework Help
Replies
1
Views
849
  • Advanced Physics Homework Help
Replies
10
Views
460
Replies
8
Views
1K
  • Introductory Physics Homework Help
Replies
11
Views
1K
  • Introductory Physics Homework Help
Replies
7
Views
1K
Replies
2
Views
227
  • Introductory Physics Homework Help
Replies
1
Views
660
Back
Top