Why Can't Quantum Numbers Include Half Integers?

Click For Summary

Discussion Overview

The discussion revolves around the nature of quantum numbers, specifically questioning why they cannot include half-integers. Participants explore this topic in the context of quantum mechanics, addressing both spin quantum numbers and solutions to the Schrödinger equation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant expresses confusion about the inability to have half-integer quantum numbers and seeks clarification.
  • Another participant suggests that if the discussion pertains to spin quantum numbers, the quantization can be derived from angular momentum commutation relations, indicating that this is a well-established aspect of quantum mechanics.
  • A further point is made regarding the solutions to the Schrödinger equation, noting that these solutions are not always quantized and depend on the Hamiltonian, which can lead to non-normalizable wave functions if certain conditions are not met.
  • It is mentioned that boundary conditions play a crucial role in enforcing discrete values for energy, ensuring that wave functions remain single-valued and normalizable.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial question regarding half-integer quantum numbers. Multiple perspectives are presented, particularly concerning the nature of quantum numbers related to spin and the solutions to the Schrödinger equation.

Contextual Notes

The discussion highlights limitations in understanding the conditions under which quantum numbers are quantized, particularly the dependence on Hamiltonians and boundary conditions. There is an acknowledgment of the complexity surrounding the quantization process without resolving the underlying issues.

HiggsBoson1
Messages
10
Reaction score
4
I don't understand why quantum numbers can not be divided into half integers and so on. The books I have read do not give clear explanations. Would anyone mind helping me understand this? Thanks!
 
Physics news on Phys.org
What quantum numbers are you talking about?

If you are talking about spin quantum numbers then any decent book on QM will prove from the angular momentum commutation relations why its quantized.

If you are talking about solutions to Schroedinger's equation then that is not always quantized, and when it is it depends entirely on the Hamiltonian:
http://www.physics.ox.ac.uk/Users/smithb/website/coursenotes/qi/QILectureNotes3.pdf

Thanks
Bill
 
  • Like
Likes   Reactions: HiggsBoson1
bhobba said:
What quantum numbers are you talking about?

If you are talking about spin quantum numbers then any decent book on QM will prove from the angular momentum commutation relations why its quantized.

If you are talking about solutions to Schroedinger's equation then that is not always quantized, and when it is it depends entirely on the Hamiltonian:
http://www.physics.ox.ac.uk/Users/smithb/website/coursenotes/qi/QILectureNotes3.pdf

Thanks
Bill
That was really helpful! Thanks a lot! :)
 
HiggsBoson1 said:
I don't understand why quantum numbers can not be divided into half integers and so on. The books I have read do not give clear explanations. Would anyone mind helping me understand this? Thanks!

The Schrödinger equation: H \psi = E \psi can be solved for arbitrary values of E, but when E is not an eigenvalue of the hamiltonian, then \psi will be unnormalizable--it will blow up at infinity, or at the origin, or somewhere. For example, a solution to the free particle Schrödinger equation with E < 0 is: \psi = e^{\hbar K x}, which corresponds to an energy of -\hbar^2 K^2/(2 m).

The differential equations for the wave function are forced to have discrete values by boundary conditions: to make sure that the wave function is single-valued in the whole domain, and to make sure that the integral of |\psi|^2 is finite.
 
  • Like
Likes   Reactions: HiggsBoson1

Similar threads

Undergrad Why is m_j not a good quantum number in strong-field Zeeman effect?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad How to understand quantum computers?
  • · Replies 9 ·
Replies
9
Views
3K
Undergrad How are good quantum numbers related to perturbation theory?
  • · Replies 18 ·
Replies
18
Views
4K
Undergrad Matrix elements of Hamiltonian include Dirac delta term
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Number of qubits needed to simulate the universe
  • · Replies 22 ·
Replies
22
Views
5K
Undergrad Determining the Number of Points in the n-Space
  • · Replies 16 ·
Replies
16
Views
2K
High School Direct sum decomposition into orthogonal subspaces
  • · Replies 2 ·
Replies
2
Views
1K
Graduate Cause of spontaneous emission?
  • · Replies 15 ·
Replies
15
Views
4K
High School Quantum Field Theory As Simple As Possible
  • · Replies 2 ·
Replies
2
Views
2K
High School Frequency of an EM wave in Classical and Quantum Physics
  • · Replies 6 ·
Replies
6
Views
2K