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Section 10.5: Classical and Quantum-mechanical Probabilities
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In the previous two Sections we discussed the energy eigenfunctions of the infinite square well. In this Section we derive the momentum-space energy eigenfunction and compare it directly to the position-space energy eigenfunction as shown in the animation. Restart.
The momentum-space energy eigenfunction is given by the Fourier transform of the position-space energy eigenfunction. Since the position-space energy eigenfunction is zero outside of the well, the Fourier transform should just involve the integral over the well:
φn(p)= (2πħ)−1/2 ∫ψn(x) exp(−ipx/ħ) dx [integral from 0 to L], (10.8)
φn(p)= −i (L/4πħ)1/2 exp[−ipL/(2ħ)] [ exp(+inπ/2) (sin(δn−)/δn− − exp(−inπ/2) (sin(δn+)/δn+ ], (10.9)
where δn+ ≡ (pL/ħ + nπ)/2 and δn− ≡ (pL/ħ − nπ)/2 . Note that the largest peaks in the momentum-space energy eigenfunction occur when δn+ or δn− are zero which corresponds to when p = ± nπħ/L. This agrees with classical expectations. The unexpected structure in the momentum-space energy eigenfunction arises because the position-space energy eigenfunction does not extend over all space, thereby complicating the results of the Fourier transform.
In the animations, you can change n and see the resulting changes in the position-space and momentum-space energy eigenfunction. The time is given in terms of the time it takes the ground-state time-dependent energy eigenfunction to return to its original phase, i.e., Δt = 1 corresponds to an elapsed time of 2πħ/E1.
Using the first check box, you can view the probability densities in position and momentum space. The probability density in momentum space is
|φn(p)|2 = (L/(4ħπ)) [sin2(δn−)/δn−2 + sin2(δn+)/δn+2 − 2 cos(nπ) (sin(δn−)sin(δn+))/(δn−δn+)]. (10.10)
In the animation you can also check the box that superimposes the normalized classical probability distributions (in pink) on the quantum-mechanical probability densities. Note that the classical position-space probability distribution is uniform over the entire well and therefore you would expect an equal likelihood of finding the classical particle anywhere in the well. The classical momentum-space probability distribution consists of two spikes at p = ± nπħ/L. They correspond to the fact that half the time the classical particle is moving to the right and half the time the classical particle is moving to the left within the well.