Commutator of the Hamiltonian with Position and Hamiltonian with Momentum

In summary, the commutator of the Hamiltonian with position and Hamiltonian with momentum is a mathematical operation that measures the non-commutativity of these two quantities. It is calculated by taking the difference between the product of the two quantities in two different orders. The commutator has physical significance in quantum mechanics, representing uncertainty and playing a crucial role in the Heisenberg uncertainty principle and time evolution of quantum systems. While it can be zero in certain cases, it is usually non-zero and is directly related to the energy of a quantum system.
  • #1
arsenalfan
1
0
To prove:

xnohou.gif

pnohou.gif


Commutator of the Hamiltonian with Position:
i have been trying to solve, but i am getting a factor of 2 in the denominator carried from p2/2m

Commutator of the Hamiltonian with Momentum:
i am not able to proceed at all...


Kindly help.. :(
 
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  • #2
Please show us your work.
 
  • #3
Post your work and take into consideration the hint to compute the commutators when applied on a wavefunction from their common dense everywhere domain.
 

1. What is the commutator of the Hamiltonian with position and Hamiltonian with momentum?

The commutator of the Hamiltonian with position and Hamiltonian with momentum is a mathematical operation that measures the degree to which these two quantities do not commute or do not have a defined order of multiplication. It is denoted by [H, x] for the commutator of Hamiltonian with position, and [H, p] for the commutator of Hamiltonian with momentum.

2. How is the commutator calculated?

The commutator is calculated by taking the difference between the product of the two quantities in two different orders. For example, the commutator of Hamiltonian with position can be calculated as [H, x] = Hx - xH, and the commutator of Hamiltonian with momentum can be calculated as [H, p] = Hp - pH.

3. What is the physical significance of the commutator of Hamiltonian with position and Hamiltonian with momentum?

The commutator of the Hamiltonian with position and Hamiltonian with momentum is a fundamental quantity in quantum mechanics. It represents the uncertainty or non-commutativity of these two quantities and plays a crucial role in the Heisenberg uncertainty principle. It also determines the time evolution of a quantum system and is a key component in understanding the behavior of quantum particles.

4. Can the commutator of Hamiltonian with position and Hamiltonian with momentum be zero?

Yes, the commutator of Hamiltonian with position and Hamiltonian with momentum can be zero in certain cases. For example, in a system with a constant potential, the commutator of these two quantities is zero. However, in most cases, the commutator is non-zero, indicating the non-commutativity and uncertainty of these quantities.

5. How does the commutator of Hamiltonian with position and Hamiltonian with momentum affect the energy of a quantum system?

The commutator of Hamiltonian with position and Hamiltonian with momentum is directly related to the energy of a quantum system. In fact, the commutator of these two quantities is equal to the negative of the system's kinetic energy. This relationship is crucial in understanding the quantization of energy in quantum mechanics and plays a significant role in various physical phenomena.

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