Commutation relations of P and H

  • Context: Graduate 
  • Thread starter Thread starter orienst
  • Start date Start date
  • Tags Tags
    Commutation Relations
Click For Summary
SUMMARY

The discussion centers on the commutation relations between the momentum operator (P) and the Hamiltonian operator (H) in quantum mechanics, specifically within the context of an infinite square well. It is established that the commutator [P, H] can be calculated, yielding the result [P, H] = -dV/dx, where V(x) represents the potential energy. The discussion emphasizes that the domain of the product operator must intersect with the domain of the Hamiltonian, specifically D(AH) ∩ D(HA) ≠ ∅, which is crucial for the validity of the commutation relations.

PREREQUISITES
  • Understanding of quantum mechanics and operators
  • Familiarity with commutation relations and their implications
  • Knowledge of the infinite square well potential
  • Basic calculus, particularly differentiation
NEXT STEPS
  • Study the implications of commutation relations in quantum mechanics
  • Learn about the properties of the infinite square well potential
  • Explore the concept of operator domains in quantum mechanics
  • Investigate the role of potential energy functions in quantum systems
USEFUL FOR

Quantum physicists, students of quantum mechanics, and anyone interested in the mathematical foundations of quantum observables and their relationships.

orienst
Messages
16
Reaction score
0
Can we always calculate the commutation relations of two observables? If so, what’s the commutator of P (momentum) and H (Hamiltonian) in infinite square well, considering that the momentum is not a conserved quantity?
 
Physics news on Phys.org
For a quantum observable, call it A, it's not important that it doesn;t commute with the Hamiltonian. It only matters that

D(AH)∩D(HA)≠\emptyset

and the domain of the product operator is the subset of D(H), such as

I(H)⊂D(A)
 
Remember that [P,.] works like a derivative for x. So, generically, for H = P^2/2m + V(x) where V is any potential, [P,H] = -dV/dx. In classical mechanics, this is the force. For infinite square potential, it gives two delta spikes at the edges of the box.
 

Similar threads

Undergrad Commutator of x and p in quantum mechanics
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad I would like some assistance working with commutators
  • · Replies 1 ·
Replies
1
Views
1K
Graduate How Does Position Interact with Spin Angular Momentum in Quantum Mechanics?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Necessary conditions for the uncertainty principle
  • · Replies 7 ·
Replies
7
Views
947
Undergrad Question about Commutators and Uncertainty
  • · Replies 17 ·
Replies
17
Views
3K
Undergrad Momentum eigenfunctions in an infinite well
  • · Replies 31 ·
2
Replies
31
Views
4K
Undergrad Time-energy uncertainty relation
  • · Replies 33 ·
2
Replies
33
Views
4K
Graduate Symmetries of particle interacting with external fields (Ballentine)
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Anti-commutation relation for quantized fields
  • · Replies 13 ·
Replies
13
Views
2K
Graduate Does x affect the value of [a^2,(a^†)^2×e^2ikx]
  • · Replies 2 ·
Replies
2
Views
2K