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Time-independent SE linear combination solution help

  1. Feb 26, 2015 #1

    Maylis

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    Gold Member

    Hello,

    I am trying to derive the TISE, but I am having many questions, and the textbook (Griffiths) does not give any adequate explanation and I have minimal access to my professor. My goal is to find ##\Psi (x,t)##. The book says the solution is

    $$ \Psi (x,t) = \sum_{n=0}^{\infty} c_{n} \psi_{n}(x) exp(\frac {-iE_{n}t}{\hbar}) $$

    So I start with the general SE (I am just taking this as a fact since the first page of Griffiths puts this equation and says its right)

    $$ i \hbar \frac {\partial \Psi}{\partial t} = - \frac {\hbar^{2}}{2m} \frac {\partial^{2} \Psi}{\partial x^{2}} + V \Psi $$

    I can separate ##\Psi (x,t) = \psi (x) f(t)## and substitute

    $$ i \hbar \frac {\partial [\psi (x) f(t)]}{\partial t} = - \frac {\hbar^{2}}{2m} \frac {\partial^{2} [\psi (x) f(t)]}{\partial x^{2}} + V [\psi (x) f(t)]$$

    Then divide by ##\psi (x) f(t)## in order to have the potential, ##V##, stand alone

    $$ i \hbar \frac {1}{f} \frac {\partial [f(t)]}{\partial t} = - \frac {\hbar^{2}}{2m} \frac {1}{\psi (x)} \frac {\partial^{2} [\psi (x)]}{\partial x^{2}} + V $$

    Now comes the first question. Griffiths says (without explanation)

    $$ i \hbar \frac {1}{f(t)} \frac {\partial [f(t)]}{\partial t} = - \frac {\hbar^{2}}{2m} \frac {1}{\psi (x)} \frac {\partial^{2} [\psi (x)]}{\partial x^{2}} + V = E $$

    Now, what exactly is ##E##? Is this "equality" actually just a definition, such that ##E## is just defined this way? Also, does ##E## stand for "energy"?

    Anyways, they go on to find ##f(t)##, which with trivial integration is exponential, and ##f(t) = exp( \frac {-iE}{\hbar} t)##

    So now I'm at this point, and I am not seeing how I will find ## \Psi (x,t) ##. I do know ## \Psi (x,t) = \psi (x) exp( \frac {-iE}{\hbar} t)## My guess is to try and substitute back into the SE

    $$ i \hbar \frac {\partial [\psi (x) exp( \frac {-iE}{\hbar} t)]}{\partial t} = - \frac {\hbar^{2}}{2m} \frac {\partial^{2} [\psi (x) exp( \frac {-iE}{\hbar} t)]}{\partial x^{2}} + V [\psi (x) exp( \frac {-iE}{\hbar} t)] $$

    Okay, well where the heck do I get this linear combination? This just got ugly, and the book doesn't show the steps.
     
    Last edited: Feb 26, 2015
  2. jcsd
  3. Feb 26, 2015 #2
    The equality in your first question comes about because you have two functions, each dependent on a different variable, which are equal. The only way to get this is if both functions are equal to something that depends on neither variable i.e. a constant. The choice of calling this constant ##E## is made a posteriori and does indeed refer to energy.

    I'm afraid I can't remember how to answer your second question, which is a shame since I only learned it last year out of the same textbook!
     
  4. Feb 28, 2015 #3

    Maylis

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    Anyone know how to make that critical step to go from the differential equation to a solution that is an infinite series?
     
  5. Mar 1, 2015 #4
    The solution constructed as an infinite series is derived from the linearity of the Schroedinger equation. If you found two solutions then any linear combination of the two solutions is a solution of the Schroedinger equation.
     
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