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Time independent Schrödinger Eqn in a harmonic potential

  1. Jul 31, 2017 #1
    1. The problem statement, all variables and given/known data
    I am currently reading a text book on solving the Schrödinger equation for the harmonic oscillator using the series method;
    $$-\frac{\hbar^{2}}{2m}\frac{\mathrm{d}^2 \psi }{\mathrm{d} x^2}+\frac{1}{2}m\omega ^{2}x^2\psi =E\psi $$

    It starts by using these two dimensionless variable substitutions (which I gather is standard practise): $$\xi \equiv \sqrt{\frac{m\omega }{\hbar}}x$$and$$K\equiv \frac{2E}{\hbar\omega }$$
    to produce the simplified equation: $$\frac{\mathrm{d} ^2\psi }{\mathrm{d} \xi ^2}=\left ( \xi ^2-K \right )\psi $$

    I cannot match the final equation using these substitutions alone. Surely there must be some adjustment for the change of variables in the derivative? Please can someone explain how to get the final (simplified) equation? (I am a newcomer to quantum mechanics!!)

    Thanks in advance.
     
  2. jcsd
  3. Aug 2, 2017 #2

    vela

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    You need to use the chain rule: ##\frac{d}{dx} = \frac{d\xi}{dx} \frac{d}{d\xi}##.
     
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