Creation and annihilation operators

Click For Summary

Discussion Overview

The discussion centers on the creation and annihilation operators in quantum mechanics, particularly in the context of one-dimensional potentials, such as the harmonic oscillator potential. Participants explore the applicability of these operators to other potentials and the mathematical and physical implications of their use.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants note that creation and annihilation operators are specifically defined for the harmonic oscillator potential, questioning why similar operators cannot be defined for other potentials.
  • Others suggest that the mathematical structure may not support the reduction of operators into commutation relations in non-harmonic potentials, implying that the simplicity of the harmonic potential is crucial for the tractability of these operators.
  • It is proposed that stepping operators can also be applied to solve the Coulomb problem and the n-dimensional harmonic oscillator, indicating that such operators exist in principle but may not always be easily expressed.
  • One participant expresses curiosity about the terminology used for excited states in different problems, specifically questioning why phonons are primarily associated with the harmonic oscillator.
  • A reference is provided regarding the stepping operators for the hydrogen atom, linking them to an SO(4) symmetry group and their role in separability in parabolic coordinates.

Areas of Agreement / Disagreement

Participants express differing views on the applicability and definition of creation and annihilation operators across various potentials. There is no consensus on why these operators are predominantly discussed in the context of the harmonic oscillator.

Contextual Notes

Some limitations include the dependence on specific mathematical structures and the unresolved nature of how these operators relate to other potentials beyond the harmonic oscillator.

LagrangeEuler
Messages
711
Reaction score
22
In one dimensional problems in QM only in case of the potential ##V(x)=\frac{m\omega^2x^2}{2}## creation and annihilation operator is defined. Why? Why we couldn't define same similar operators in cases of other potentials?
 
Physics news on Phys.org
LagrangeEuler said:
In one dimensional problems in QM only in case of the potential ##V(x)=\frac{m\omega^2x^2}{2}## creation and annihilation operator is defined. Why? Why we couldn't define same similar operators in cases of other potentials?

I guess mathematics wouldn't work and you would not be able to reduce those operators into entities with commutation operators leading to simple algebraic relations. Plus this harmonic potential corresponds to free fields, and this "freeness" i guess is the physical reason why things are simple and tractable and meaningful.
 
Stepping operators can also be used to solve the Coulomb problem. Also the n-dimensional harmonic oscillator. In principle such operators always exist, but only in a few problems are they simple enough to write down in useful form.
 
Bill_K said:
Stepping operators can also be used to solve the Coulomb problem. Also the n-dimensional harmonic oscillator. In principle such operators always exist, but only in a few problems are they simple enough to write down in useful form.


I suppose that. That they always could be written. But why we that speak about phonons only in that problem. Why we don't give a name of excited states in some other problem.

Stepping operators can also be used to solve the Coulomb problem.
Do you have reference for this?
 
The stepping operators for the H atom form the generators of an SO(4) symmetry group, and are also related to its separability in parabolic coordinates. Here's a paper that talks about it.
 
Also see the reference given in this earlier thread, post #2, courtesy of dextercioby.

The book itself is available online http://www.scribd.com/doc/22703322/Fitts-D-D-Principles-of-Quantum-Mechanics-As-Applied-to-Chemistry-and-Chemical-Physics.
 
Last edited:

Similar threads

Graduate Creation and annihilation operator
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Confusion about creation/annihilation operators with interaction
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Creation/annihilation operators question
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Explicit form of annihilation and creation operators for Dirac field
  • · Replies 6 ·
Replies
6
Views
3K
Graduate Annihilation creation operators in case of half H.oscillator
  • · Replies 14 ·
Replies
14
Views
3K
Graduate Creation/annihilation operators and trigonometric functions
  • · Replies 9 ·
Replies
9
Views
3K
Graduate Quantum fields and the harmonic oscillator
  • · Replies 4 ·
Replies
4
Views
2K
Graduate The physical derivation of annihilation operator?
  • · Replies 13 ·
Replies
13
Views
5K
Graduate Expectation value in Heisenberg picture: creation and annihilation
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Question about creation and annihilation operators?
  • · Replies 6 ·
Replies
6
Views
5K