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Wave-functions help! Hamiltonian and Free space?!

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!
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))
= ((ħ2 k2)/2m) cos(kx)

so energy is (ħ2 k2)/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
(edited 6 years ago)
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PencilPot!
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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))
= ((ħ2 k2)/2m) cos(kx)

so energy is (ħ2 k2)/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 :smile:

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