# Ground state energy of 5 identical spin 1/2 particle

• Sushmita
In summary, the ground state energy of 5 identical spin 1/2 particles in a one-dimensional simple harmonic oscillator potential of frequency ω is (13/2) ħω. This is because the particles are fermions, so they cannot occupy the same state. Each particle will have a different state and energy, corresponding to n= 0,1,2,3,4. Therefore, the total energy is calculated as 2(0+1/2)ħω + 2(1+1/2)ħω + 1(2+1/2)ħω = (13/2)ħω.

## Homework Statement

The ground state energy of 5 identical spin 1/2 particles which are subject to a one dimensional simple harkonic oscillator potential of frequency ω is
(A) (15/2) ħω
(B) (13/2) ħω
(C) (1/2) ħω
(D) 5ħω

## Homework Equations

Energy of a simple harmonic oscillator potential is
En = (n+½) ħω

## The Attempt at a Solution

Since the particles have 1/2 spin so they are fermions. So any two particpes cannot be in the same state. So every particle will have a different state and hence a different energy corresponding to that state.
To calculate the ground state so each of the 5 particles will have energy corresponding to n= 0,1,2,3,4

(1/2)ħω
(3/2)ħω
(5/2)ħω
(7/2)ħω
(9/2)ħω

So total energy of the particle is = (25/2)ħω

But this is not the answer. Can you let me what am I doing wrong?

Sushmita said:
So any two particpes cannot be in the same state. So every particle will have a different state and hence a different energy corresponding to that state.
There can be more than one state with the same energy.

TSny said:
There can be more than one state with the same energy.
But this is a one dimensional potention. There is no degeneracy.

A possible state of a spin 1/2 particle is to have an energy of ħω/2 with its spin "up". But that's not the only way the particle could have an energy of ħω/2.

Last edited:
TSny said:
A possible state of a spin 1/2 particle is to have an energy of ħω/2 with its spin "up". But that's not the only way the particle could have an energy of ħω/2.
Okay. i get it now. Thanks a lot.

Sushmita said:

## Homework Statement

The ground state energy of 5 identical spin 1/2 particles which are subject to a one dimensional simple harkonic oscillator potential of frequency ω is
(A) (15/2) ħω
(B) (13/2) ħω
(C) (1/2) ħω
(D) 5ħω

## Homework Equations

Energy of a simple harmonic oscillator potential is
En = (n+½) ħω

## The Attempt at a Solution

Since the particles have 1/2 spin so they are fermions. So any two particpes cannot be in the same state. So every particle will have a different state and hence a different energy corresponding to that state.
To calculate the ground state so each of the 5 particles will have energy corresponding to n= 0,1,2,3,4

(1/2)ħω
(3/2)ħω
(5/2)ħω
(7/2)ħω
(9/2)ħω

So total energy of the particle is = (25/2)ħω

But this is not the answer. Can you let me what am I doing wrong?
Here filling of electron will be 2,2,1=5
So E=2(0+1/2)hcutw +2(1+1/2)hcutw+1(2+1/2)hcutw
E= 13/2hcutw