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

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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!
    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
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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))
    = ((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
 
 
 
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