Consider an unormalized wavefunction of a rotor at $t = 0$, a combination of $n=0$ and $n=2$ states:

$$\psi(\phi) = 3 - 2 \cos (2\phi).$$

Find the probability distribution in angle.

The book simply takes probability as

$$\lvert\psi(\phi)\rvert^2 = \lvert 3 - 2\cos(2\phi)\rvert^2 = 9 + 4\cos^2(2\phi) - 12\cos(2\phi).$$

I have found the normalized wavefunction to be:

$$\psi(\phi) = \frac{3}{\sqrt {11}}\lvert 0\rangle - \frac{2}{\sqrt {22}}\lvert 2\rangle.$$

Do I use probability as $\left\lvert\frac{3}{\sqrt {11}} - \frac{2}{\sqrt {22}}\cos(2\phi)\right\rvert^2$ ? or the one in the book?

  • 3
    $\begingroup$ Probabilities must add up to 1 $\endgroup$ – Floris Apr 22 '14 at 3:33
  • $\begingroup$ which one do i use? $\endgroup$ – user44840 Apr 22 '14 at 4:27

For valid probabilities, use the normalized wave function. If all you care about is the shape of the probability distribution (i.e., which values are more likely than others, and relatively speaking how much more likely are they?), you can save time by working with the un-normalized function.


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