You are Here: Home

# Wave-functions help! Hamiltonian and Free space?! watch

1. 1) Ψ1 = sin(𝑘𝑥)
2) Ψ2 = e^𝑖𝑘𝑥 = 𝑐𝑜𝑠(𝑘𝑥) + 𝑖sin(𝑘𝑥)

For each wave function, show that they are eigenfuntions of the hamiltonian in
i) Free Space (V(x)=0).
ii) A 'flat' potential (V(x) = V)

In each case, what is the kinetic and total energy?

I've done all of it, but how do I show in each case, what is the kinetic and total energy?

Thanks!
2. (Original post by PencilPot!)
1) Ψ1 = sin(𝑘𝑥)
2) Ψ2 = e^𝑖𝑘𝑥 = 𝑐𝑜𝑠(𝑘𝑥) + 𝑖sin(𝑘𝑥)

For each wave function, show that they are eigenfuntions of the hamiltonian in
i) Free Space (V(x)=0).
ii) A 'flat' potential (V(x) = V)

In each case, what is the kinetic and total energy?

I've done all of it, but how do I show in each case, what is the kinetic and total energy?

Thanks!
The Hamiltonian is the energy operator, so for a wavefunction Ψ that is an eigenfunction of the hamiltonian

Ĥ Ψ = E Ψ

so total energy is the eigenvalue you get, which will be whatever is multiplying the original wavefunction.

An example (assuming V(x)=0)

Ψ=cos(kx)

Ĥ Ψ = -(ħ2/2m)d2/dx2 (cos(kx))
= ((ħ2k2)/2m) cos(kx)

so energy is (ħ2k2)/2m

Total energy= kinetic energy + potential energy

for case one in the question, we have no potential energy so total energy=kinetic energy

for case two potential energy is a constant, V, now total energy= kinetic energy + V
3. (Original post by MexicanKeith)
The Hamiltonian is the energy operator, so for a wavefunction Ψ that is an eigenfunction of the hamiltonian

Ĥ Ψ = E Ψ

so total energy is the eigenvalue you get, which will be whatever is multiplying the original wavefunction.

An example (assuming V(x)=0)

Ψ=cos(kx)

Ĥ Ψ = -(ħ2/2m)d2/dx2 (cos(kx))
= ((ħ2k2)/2m) cos(kx)

so energy is (ħ2k2)/2m

Total energy= kinetic energy + potential energy

for case one in the question, we have no potential energy so total energy=kinetic energy

for case two potential energy is a constant, V, now total energy= kinetic energy + V
Thank you!
I got that in the end

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: October 24, 2017
The home of Results and Clearing

### 2,241

people online now

### 1,567,000

students helped last year
Today on TSR

### University open days

1. Sheffield Hallam University
Tue, 21 Aug '18
2. Bournemouth University
Wed, 22 Aug '18
3. University of Buckingham
Thu, 23 Aug '18
Poll
Useful resources

Can you help? Study Help unanswered threadsStudy help rules and posting guidelines

## Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE