1d potential and switching between operators

In summary, the group is stuck on a homework problem because the lecturer has refused to provide help, encouraging them to work together as a group. The attempt at a solution involved substituting for p and using commutation relationships, but this did not lead to a solution. Another attempt was made by substituting for H, but the resulting brackets were not equal. The suggestion was made to calculate [x,H] and equate it to <n|[x,H]|n'> to solve the problem.
  • #1
Brewer
212
0

Homework Statement


q1.jpg



Homework Equations





The Attempt at a Solution


As a group we're stuck on this as a result of the lecturer saying that he wouldn't help us because we should work as a group and find other ways other than asking him about it. Which is fair enough - but none of us understand what's going on!

So far I have tried substituting in for p in the middle of the bracket with -ihd/dx and then used the commutation relationship to try substituting ih for xp - px, but this seems to leave me being nowhere.

I also tried starting with the bracket having H in the middle and substituting for that - this kind of left me with a bracket with p in the middle and one with x in the middle, but whilst this kind of left things in a format that almost looked correct the two brackets were not equal to one another, only additive.

Could anyone suggest a starting point to look at or just any other helpful information that I may have missed when rereading my notes.
 
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  • #2
Well, have you calculated [x,H]?

Then you can calculate <n|[x,H]|n'> in two different ways: equating the two answers gives you the desired result.
 
  • #3
No I hadn't - I thought that was to do with the second part of the question.

I shall give that a go now!
 

Related to 1d potential and switching between operators

1. What is 1d potential?

1d potential refers to the potential energy of a system in one-dimensional space. It is a measure of the energy that a particle has based on its position in a one-dimensional system. This potential energy can be represented by a potential energy curve, which shows the relationship between the particle's position and its potential energy.

2. What is the significance of switching between operators in 1d potential?

In 1d potential, switching between operators refers to changing the mathematical operator used to describe the system. This is significant because different operators can provide different information about the system, such as the particle's position, momentum, or energy. By switching between operators, scientists can gain a more comprehensive understanding of the system and its behavior.

3. What are the common operators used in 1d potential?

The most common operators used in 1d potential are the position operator, momentum operator, and Hamiltonian operator. The position operator, denoted by x, represents the particle's position in the system. The momentum operator, denoted by p, represents the particle's momentum. The Hamiltonian operator, denoted by H, represents the total energy of the system.

4. How does the potential energy affect the switching between operators in 1d potential?

The potential energy of a system directly affects the behavior of the particle and, therefore, the switching between operators in 1d potential. Different potentials, such as harmonic, square well, or Coulomb, will result in different energy levels and, consequently, different behaviors of the particle. This, in turn, will affect the results obtained by switching between operators.

5. How is 1d potential and switching between operators used in real-world applications?

1d potential and switching between operators have many real-world applications, particularly in the field of quantum mechanics. They are used to understand the behavior of particles in one-dimensional systems, such as atoms, molecules, and semiconductors. This knowledge is crucial in the development of new technologies, such as nanotechnology and quantum computing.

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