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

## 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?

TSny
Homework Helper
Gold Member
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.

There can be more than one state with the same energy.
But this is a one dimentional potention. There is no degeneracy.

TSny
Homework Helper
Gold Member
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:
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.

## 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