Operator Theory Problem on Momentum Operator (QM)

In summary, the conversation discusses how to prove a given equation involving operators and their corresponding functions. The suggested approaches include expanding the function in a power series and considering the case in momentum-space rather than position-space.
  • #1
6
0

Homework Statement



Given the operators [itex]\hat{x}=x\cdot[/itex] and [itex]\hat{p}=-i\hbar \frac{d}{dx}[/itex], prove that:

[itex][\hat{x}, g(\hat{p})]=i\hbar \frac{dg}{d\hat p}[/itex]

Homework Equations



None.

The Attempt at a Solution



I have very little idea on how to begin this problem, but I don't want a solution, I simply want a hint in the right direction.

Thanks, mates.
 
Last edited:
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  • #2
LolWolf said:

Homework Statement



Given the operators [itex]\hat{x}=x\cdot[/itex] and [itex]\hat{p}=-i\hbar \frac{d}{dx}[/itex], prove that:

[itex][\hat{x}, g(\hat{p})]=i\hbar \frac{dg}{d\hat p}[/itex]

Homework Equations



None.

The Attempt at a Solution



I have very little idea on how to begin this problem, but I don't want a solution, I simply want a hint in the right direction.

Thanks, mates.

Expand ##g(p)## in a power series. What's ##[x,p^n]##?
 
  • #3
Actually, I realized it was even easier than that, but thank you!

Consider the case in momentum-space rather than position-space, and this reduces nicely using elementary operations.
 
  • #4
LolWolf said:
Actually, I realized it was even easier than that, but thank you!

Consider the case in momentum-space rather than position-space, and this reduces nicely using elementary operations.

Sure, that works also. x is a differentiation operator in p space.
 
  • #5


I would start by reviewing the definitions and properties of operators in quantum mechanics, specifically the commutator and how it relates to the uncertainty principle. Then, I would consider how the momentum operator \hat{p} can be written in terms of the position operator \hat{x} and the derivative operator d/dx. From there, I would try to apply the commutator relationship to the given operators and use properties of derivatives to simplify the expression. It may also be helpful to consider the physical meaning behind the commutator and how it relates to the measurement of momentum.
 

1. What is the Operator Theory Problem on Momentum Operator in Quantum Mechanics (QM)?

The Operator Theory Problem on Momentum Operator in Quantum Mechanics (QM) is a fundamental concept in quantum mechanics which relates to the measurement of momentum of a particle. It is based on the Heisenberg uncertainty principle which states that the position and momentum of a particle cannot be simultaneously measured with complete precision.

2. How is the momentum operator defined in QM?

In QM, the momentum operator is defined as the operator which represents the physical quantity of momentum in a quantum system. It is represented by the symbol p and is mathematically defined as -iħ(d/dx), where i is the imaginary unit and ħ is the reduced Planck's constant.

3. What is the significance of the Operator Theory Problem on Momentum Operator in QM?

The Operator Theory Problem on Momentum Operator is important in QM as it helps us understand the behavior of particles at the quantum level. It allows us to make predictions about the momentum of a particle and how it may change over time.

4. How is the momentum operator used in QM calculations?

The momentum operator is used in QM calculations to determine the momentum of a particle in a given quantum state. It is also used to calculate the expectation value of momentum, which is the average value that would be obtained from multiple measurements of a particle's momentum.

5. Can the momentum operator be generalized to higher dimensions?

Yes, the momentum operator can be generalized to higher dimensions in QM. In one dimension, it is represented by -iħ(d/dx). In three dimensions, it is represented by -iħ(∇), where ∇ is the gradient operator. This allows for the calculation of momentum in multiple directions in three-dimensional space.

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