Why is the ground state eigenfunction in a symmetric hamiltonian also symmetric?

Click For Summary
SUMMARY

The ground state eigenfunction in a quantum mechanical system is symmetric under coordinate inversion only if the Hamiltonian is symmetric. In systems with symmetric Hamiltonians, all eigenfunctions are classified as either even (symmetric) or odd (antisymmetric). Antisymmetric wavefunctions contain nodes, which disqualifies them from being the ground state. Consequently, the ground state eigenfunction must be symmetric when the Hamiltonian exhibits symmetry.

PREREQUISITES
  • Understanding of quantum mechanics principles
  • Familiarity with Hamiltonian operators
  • Knowledge of eigenfunctions and eigenvalues
  • Concept of symmetry in physical systems
NEXT STEPS
  • Study the properties of symmetric Hamiltonians in quantum mechanics
  • Explore the implications of eigenfunction symmetry on quantum states
  • Learn about nodes in wavefunctions and their significance
  • Investigate examples of quantum systems with symmetric and antisymmetric states
USEFUL FOR

Quantum physicists, students of quantum mechanics, and researchers studying the properties of Hamiltonians and eigenfunctions.

njoshi3
Messages
7
Reaction score
0
Hi,
Why is that, the ground state eigenfunction in ANY quantum mechanical system is symmetric under inversion of co-ordinates?
 
Physics news on Phys.org
Originally posted by njoshi3
Hi,
Why is that, the ground state eigenfunction in ANY quantum mechanical system is symmetric under inversion of co-ordinates?

well first of all, this isn t true. the groundstate is only symmetric if the hamiltonian is also symmetric.

so let s assume that you had asked this question about any quantum system with a symmetric hamiltonian.

when there is a symmetric hamiltonian, all eigenfuctions must be either even or odd. that is, they must be either symmetric or antisymmetric.

an antisymmetric wavefunction necessarily passes through zero. therefore it has a node, therefore it cannot be the ground state.

therefore the ground state must be symmetric.
 

Similar threads

Graduate Exact symmetry, quantum states, and symmetric dynamics
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Representation of Spin 1/2 quantum state
  • · Replies 61 ·
3
Replies
61
Views
7K
Undergrad Understanding the dynamics of a perturbed quantum harmonic oscillator
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Ground state calculation with a time averaged wave function
  • · Replies 2 ·
Replies
2
Views
2K
Graduate The half harmonic oscillator's ground state wave function
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Quantum statistical canonical formalism to find ground state at T
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Does the quantum space of states have countable or uncountable basis?
  • · Replies 67 ·
3
Replies
67
Views
8K
Graduate Understanding how quantum annealing solves QUBO problems
  • · Replies 1 ·
Replies
1
Views
2K
High School Extremely elementary questions about QM
  • · Replies 24 ·
Replies
24
Views
3K
Undergrad Confusion with superposition of states
  • · Replies 2 ·
Replies
2
Views
2K