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    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!
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    (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?
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    (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]
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    (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?
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    (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?
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    (Original post by PencilPot!)
    1/2?
    V(x) is the potential energy operator so when it is zero the potential energy will be...
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    (Original post by alow)
    V(x) is the potential energy operator so when it is zero the potential energy will be...
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    (Original post by PencilPot!)
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    Yep.
 
 
 
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