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Math Help - Intro to PDE's, Schrodinger's Equation

  1. #1
    Super Member Aryth's Avatar
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    Intro to PDE's, Schrodinger's Equation

    (a) Using the separation of variables technique, and letting the separation constant be denoted E (which turns out to be the total energy of the particle), show that the resulting differential equation that is independent of time—the so-called time-independent Schroedinger equation (TISE)—has the (1-D) form

    -\frac{\hbar}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi(x) = E\psi{x}

    where we have set \Psi(x,t) = \psi(x) T(t).

    This I have done this already... And I've narrowed down to these two differential equations:

    The one above

    i\hbar \frac{dT}{dt} = ET(t)

    (b) Also, show that the total wavefunction in this case has the form

    \Psi(x,t) = \psi(x) e^{-i\frac{E}{\hbar}t}

    Here's what I did:

    i\hbar \frac{dT}{dt} = ET(t)

    \frac{dT}{T} = \frac{E}{i\hbar}dt

    ln|T| = \frac{Et}{i\hbar} + C

    T = e^{\frac{Et}{i\hbar} + C}

    T = Ae^{\frac{Et}{i\hbar}}

    This looks only similar to the actual solution for T(t)... What am I doing wrong?
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  2. #2
    MHF Contributor
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    Quote Originally Posted by Aryth View Post
    (a) Using the separation of variables technique, and letting the separation constant be denoted E (which turns out to be the total energy of the particle), show that the resulting differential equation that is independent of time—the so-called time-independent Schroedinger equation (TISE)—has the (1-D) form

    -\frac{\hbar}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi(x) = E\psi{x}

    where we have set \Psi(x,t) = \psi(x) T(t).

    This I have done this already... And I've narrowed down to these two differential equations:

    The one above

    i\hbar \frac{dT}{dt} = ET(t)

    (b) Also, show that the total wavefunction in this case has the form

    \Psi(x,t) = \psi(x) e^{-i\frac{E}{\hbar}t}

    Here's what I did:

    i\hbar \frac{dT}{dt} = ET(t)

    \frac{dT}{T} = \frac{E}{i\hbar}dt

    ln|T| = \frac{Et}{i\hbar} + C

    T = e^{\frac{Et}{i\hbar} + C}

    T = Ae^{\frac{Et}{i\hbar}}

    This looks only similar to the actual solution for T(t)... What am I doing wrong?
    Note \frac{1}{i} = - i and absorb A into \psi(x).
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