Bra-kets and operator formalism in QM - Expectation values of momentum

In summary, the person has solved a set of problems and is asking for confirmation on the accuracy of their solutions. They have provided all the questions and answers, as well as the resources they used. They also mention a small mistake in their attempt at a solution. The person assures that, with careful manipulation of operators, the problem is easy to solve.
  • #1
Dixanadu
254
2

Homework Statement


sup guys!

I think I've solved this set of problems, but I was just wondering if I've done it right - I have no way to tell. I'll put all the questions and answers here - plus the stuff I used. So could you please tell me if there's any mistakes?
Here it is - using Word as always!
http://imageshack.com/a/img4/750/6i29.png


Homework Equations



http://imageshack.com/a/img266/4082/58m3.jpg

BTW - I just realized I put in the transpose symbol instead of the dagger - i meant the dagger for the raising operator!

The Attempt at a Solution


http://imageshack.com/a/img546/7347/1oty.jpg
 
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  • #2
My solutions are here, have a look..
 

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  • #3
If you are careful with the manipulations of the operators, it's an easy problem.
 

1. What is the "bra-ket" notation in quantum mechanics?

The "bra-ket" notation, also known as Dirac notation, is a mathematical notation used to represent quantum states and operations in quantum mechanics. It consists of a left angle bracket (bra) and a right angle bracket (ket) enclosing a mathematical expression, usually a vector or an operator.

2. How is the momentum operator represented in the bra-ket notation?

In the bra-ket notation, the momentum operator is represented as p̂. It is defined as the operator that, when applied to a state vector, gives the momentum of the system as an eigenvalue. In one dimension, it is written as p̂ = -iħ(d/dx), where i is the imaginary unit and ħ is the reduced Planck's constant.

3. What is the expectation value of momentum in quantum mechanics?

The expectation value of momentum is the average value of the momentum of a particle in a given quantum state. It is calculated as the inner product of the wave function with the momentum operator. In the bra-ket notation, the expectation value of momentum is written as ψp̂>, where ψ is the wave function and p̂ is the momentum operator.

4. How do you calculate the expectation value of momentum for a particle in a specific state?

To calculate the expectation value of momentum for a particle in a specific state, you first need to find the wave function for that state. Then, you take the inner product of the wave function with the momentum operator, ψp̂>. This will give you the expectation value of momentum for that state.

5. What is the physical significance of the expectation value of momentum?

The expectation value of momentum represents the most probable value of the momentum that a particle will have when it is measured. It also gives information about the spread or uncertainty in the momentum of the particle. In quantum mechanics, measurements of physical quantities are probabilistic, so the expectation value of momentum is an important concept in understanding the behavior of particles at the quantum level.

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