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I can't seem to figure out how to derive the coordinate representation of the $a_+$ ladder operator in quantum mechanics.

I know that $a_-$ is $\sqrt{\frac{1}{2mwh}} (mwx + i\dot{p}) $ in which where I can substitute $i\dot{p}$ to make the representation:

$$\sqrt{\frac{1}{2mwh}} (mwx + \hbar \frac{d}{dx})$$

I ultimately want to solve for the differential equation using coordinate representation of the ladder operator given a condition.

Would the coordinate representation be:

$$\sqrt{\frac{1}{2mwh}} (mwx - \hbar \frac{d}{dx})$$

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up vote 2 down vote accepted

The most blatant hint is that the lowering operator is called $a$ and the raising operator is called $a^\dagger$, equivalently $a_-$ and $a_+$ in your terminology.

So, yes... your guess is correct. You might find the following wiki pages useful.

  1. https://en.wikipedia.org/wiki/Creation_and_annihilation_operators
  2. https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator#Ladder_operator_method
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