What is the position operator in the momentum basis for a given momentum value?

In summary, the conversation discusses the need to prove the position operator in the momentum basis p for p', and the use of the hermitian property of the operator x to do so. The conversation also discusses the use of hints and integration to find the desired result.
  • #1
novop
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0

Homework Statement



I need to prove that, [tex]<p'|\hat{x}p> = i\hbar\frac{d}{dp'}\delta{p-p'}[/tex]

i.e. find the position operator in the momentum basis p for p'...

It's easy to prove that [tex]<x'|\hat{x}x> = <\hat{x}x'|x> = x'<x'|x> = x'\delta{x-x'}[/tex]
(position operator in position basis for x')
since I can use the fact that the operator x is hermitian. But what about for the first problem? Any hints?
 
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  • #2
<p'|X|p> = int_x dx <p'|X|x><x|p> = int_x dx x <p'|x> <x|p> = int_x dx x (1/√2πhbar) e-ixp/hbar <x|p> =-hbar/i int_x dx ∂/∂p' <p'|x> <x|p> = ihbar ∂/∂p' ... = what you need.
 
  • #3
Great. Thanks so much.
 

1. What are operators in switching basis?

Operators in switching basis refer to the mathematical operations or functions used to manipulate the basis states of a quantum system. These operations can include rotations, translations, and reflections.

2. How do operators affect quantum states?

Operators act on the quantum states of a system to produce new states, often with different properties. They can change the energy level, spin, or other characteristics of a quantum state.

3. What is the significance of the switching basis in quantum computing?

The switching basis is a set of basis states that are used to represent quantum information in a quantum computer. By manipulating these states using operators, we can perform calculations and solve problems that are impossible to solve with classical computers.

4. How do we choose the appropriate operators for a specific problem?

The choice of operators depends on the specific problem we are trying to solve. We need to understand the properties of the problem and how the operators will affect the quantum states to determine the most effective operators to use.

5. Can operators be combined or applied in sequence?

Yes, operators can be combined or applied in sequence to perform more complex operations. This is known as operator algebra and is an important aspect of quantum computing. By combining operators, we can create new operations that can help us solve more complicated problems.

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