Commutators, operators and eigenvalues

In summary, "Commutators, operators and eigenvalues" explores the fundamental concepts in quantum mechanics related to operators, which represent physical observables. It discusses how commutators, defined as the difference between the product of two operators in different orders, provide insights into the relationship between observables and their physical implications, such as uncertainty principles. The text also delves into eigenvalues and eigenvectors, illustrating how they relate to the measurement outcomes of observables and the state of a quantum system. Overall, it highlights the mathematical framework that underpins quantum theory and its implications for understanding physical phenomena.
  • #1
dyn
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Hi
I just wanted to check my understanding of something which has come up when first studying path integrals in QM. If x and px are operators then [ x , px ] = iħ but if x and px operate on states to produce eigenvalues then the eigenvalues x and px commute because they are just numbers. Is that correct ?
Thanks
 
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  • #2
Operators ##x## and ##p_x## don't have eigenstates in common.
 
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  • #3
Yes . i know that but x can act on a position eigenstate and p can act on a momentum eigenstate. I just want to check that once the operators act on states to produce eigenvalues , that eigenvalues commute because they are just numbers ?
 
  • #4
dyn said:
the eigenvalues x and px commute because they are just numbers. Is that correct ?
No, it makes no sense, because you don't multiply eigenvalues in QM. You could of course measure the position of one quantum system and the momentum of another and then multiply the results you get, but what sense would it make?
 
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  • #5
dyn said:
something which has come up when first studying path integrals in QM
I'm struggling to see how the question you are asking could arise in that context. Can you give a reference?
 
  • #6
dyn said:
. I just want to check that once the operators act on states to produce eigenvalues , that eigenvalues commute because they are just numbers ?
Numbers and operators commute. It does not matter from where the numbers come. 2024 commutes with operators. We do not care whether it is a measured value of position of some state or new calendar year. Happy new year!
 
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  • #7
PeterDonis said:
I'm struggling to see how the question you are asking could arise in that context. Can you give a reference?
The derivation of the path integral in QM on P211-212 of "QFT for the gifted amateur" by Lancaster & Blundell
 
  • #8
dyn said:
The derivation of the path integral in QM on P211-212 of "QFT for the gifted amateur" by Lancaster & Blundell
Yes, that is correct. All the ##p##'s and ##q_n##'s here,

1704165225828.png


are just real numbers and of course commute.
 
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FAQ: Commutators, operators and eigenvalues

What is a commutator in quantum mechanics?

A commutator in quantum mechanics is an operator that measures the difference between the sequential application of two operators. Mathematically, for two operators \( \hat{A} \) and \( \hat{B} \), the commutator is defined as \( [\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A} \). Commutators are crucial in determining the uncertainty relations and the compatibility of measurements.

Why are commutators important in quantum mechanics?

Commutators are important because they provide information about the simultaneous measurability of observables. If the commutator of two operators is zero, the observables they represent can be measured simultaneously with arbitrary precision. Non-zero commutators indicate a fundamental limit to the precision with which the corresponding observables can be known simultaneously, as exemplified by the Heisenberg uncertainty principle.

What is an operator in the context of quantum mechanics?

An operator in quantum mechanics is a mathematical entity that acts on the wavefunctions of a quantum system to extract physical information. Operators correspond to observable quantities such as position, momentum, and energy. For example, the Hamiltonian operator corresponds to the total energy of the system, and its eigenvalues represent possible energy levels.

What are eigenvalues and eigenvectors in quantum mechanics?

In quantum mechanics, eigenvalues and eigenvectors arise from the application of an operator to a wavefunction. If \( \hat{O} \) is an operator and \( \psi \) is a wavefunction such that \( \hat{O}\psi = \lambda\psi \), then \( \lambda \) is called an eigenvalue and \( \psi \) is the corresponding eigenvector (or eigenfunction). Eigenvalues represent measurable quantities, and eigenvectors represent the states of the system associated with these measurements.

How do commutators relate to the Heisenberg uncertainty principle?

The Heisenberg uncertainty principle is fundamentally related to the commutator of two operators. For two observables represented by operators \( \hat{A} \) and \( \hat{B} \), the uncertainty principle states that the product of their uncertainties is bounded by the magnitude of their commutator: \( \Delta A \Delta B \geq \frac{1}{2}|\langle[\hat{A}, \hat{B}]\rangle| \). This relationship shows that non-commuting operators lead to intrinsic uncertainties in the simultaneous measurement of the corresponding observables.

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