Turn on thread page Beta
    • Thread Starter
    Offline

    7
    ReputationRep:
    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!
    Offline

    19
    ReputationRep:
    (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!
    What's the Hamiltonian operator in each case?
    • Thread Starter
    Offline

    7
    ReputationRep:
    (Original post by alow)
    What's the Hamiltonian operator in each case?
    i) ħ^2k^2/2m sin(kx)

    ii) sin(kx) [ħ)^2k^2/2m + V]

    i) -ħ^2 i^2 k^2/2m e^ikx

    ii) e^ikx [-ħ^2 i^2 k^2/2m + V]
    Offline

    19
    ReputationRep:
    (Original post by PencilPot!)
    i) ħ^2k^2/2m sin(kx)

    ii) sin(kx) [ħ)^2k^2/2m + V]

    i) -ħ^2 i^2 k^2/2m e^ikx

    ii) e^ikx [-ħ^2 i^2 k^2/2m + V]
    Ignoring constants, the Scrodinger equation will be:

    \hat{H} \Psi_n = \left[ -\dfrac12 \dfrac{\text{d}^2}{\text{d}x^2} + V(x) \right] \Psi_n = \text{E}_n \Psi_n

    - \dfrac12 \dfrac{\text{d}^2}{\text{d}x^2} is the kinetic energy operator part of the Hamiltonian.

    So for V=0, how much potential energy is there?
    • Thread Starter
    Offline

    7
    ReputationRep:
    (Original post by alow)
    Ignoring constants, the Scrodinger equation will be:

    \hat{H} \Psi_n = \left[ -\dfrac12 \dfrac{\text{d}^2}{\text{d}x^2} + V(x) \right] \Psi_n = \text{E}_n \Psi_n

    - \dfrac12 \dfrac{\text{d}^2}{\text{d}x^2} is the kinetic energy operator part of the Hamiltonian.

    So for V=0, how much potential energy is there?
    1/2?
    Offline

    19
    ReputationRep:
    (Original post by PencilPot!)
    1/2?
    V(x) is the potential energy operator so when it is zero the potential energy will be...
    • Thread Starter
    Offline

    7
    ReputationRep:
    (Original post by alow)
    V(x) is the potential energy operator so when it is zero the potential energy will be...
    0
    Offline

    19
    ReputationRep:
    (Original post by PencilPot!)
    0
    Yep.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: October 22, 2017
Poll
Do you think parents should charge rent?

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

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.