Sum of two planewaves using momentum operator

In summary, the conversation discusses a problem in a quantum physics course involving using the momentum operator to find an expression for the sum of two planewaves with the same kinetic energy in one dimension. The momentum operator is identified as (h_bar/i)(d/dx) and the two planewaves are given as exp(kx-wt) and exp(kx+wt). The goal is to find an expression on the form of the Schrodinger equation by applying the momentum operator twice. The answer is a standing wave with two possible momentum values, +hk or -hk.
  • #1
DevoBoy
8
0
Hi,

I'm baffled by a problem in a quantum physics course I'm taking.

Problem: "Use the momentum operator to find an expression for the sum of two planewaves moving in opposite directions. Both planewaves have the same kinetic energy."

It's in one dimension only.

I know the momentum operator is (h_bar/i)(d/dx), and one planewave is exp(kx-wt). The other is exp(kx+wt) ??

My main problem is that I don't know quite where to start. :rofl: How should I use the momentum operator? The answer should be on the same form as a solution to the Scrodinger equation.. :uhh:
 
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  • #2
Try writing the expression expi(kx-wt) + exp-i(kx+wt). The idea is that the wavenumber changes sign, not the frequency. This simplifies to 2coskx exp-iwt. Now apply the momentum operator twice. You'll see that p^2=(hk)^2. This means that p=+/-hk.

This is a standing wave, which describes a particle moving in either +x or -x. Supposedly, a measurement of the momentum will produce either +hk or -hk.
 
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1. What is the momentum operator in quantum mechanics?

The momentum operator in quantum mechanics is a mathematical operator that describes the momentum of a particle in a quantum system. It is represented by the symbol p and is defined as the product of the particle's mass and its velocity.

2. How is the sum of two planewaves calculated using the momentum operator?

The sum of two planewaves using the momentum operator is calculated by first expressing the planewaves as a combination of momentum eigenstates. Then, the momentum operator is applied to each eigenstate and multiplied by its corresponding coefficient. Finally, the results are summed together to get the total momentum.

3. What is the significance of calculating the sum of two planewaves using the momentum operator?

Calculating the sum of two planewaves using the momentum operator allows us to determine the total momentum of a system, which is a fundamental quantity in quantum mechanics. It also allows us to understand the behavior of particles and their interactions within a quantum system.

4. Can the sum of two planewaves using the momentum operator be used to predict the motion of a particle?

No, the sum of two planewaves using the momentum operator alone cannot predict the motion of a particle. Other factors such as the particle's initial position, potential energy, and the Schrödinger equation must also be considered.

5. Are there any limitations to calculating the sum of two planewaves using the momentum operator?

Yes, there are limitations to this calculation. The momentum operator is only applicable to particles in a quantum system and may not accurately describe the behavior of macroscopic objects. Additionally, the calculation assumes that the system is in a stationary state, which may not always be the case.

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