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In the previous two Sections we discussed the energy eigenstates of the infinite square well by deriving the position-space wave function. In this Section we derive the momentum-space wave function and compare it directly to the position-space wave function as shown in the animation. Restart.
The momentum-space wave function is given by the Fourier transform of the position-space wave function. Since the position-space wave function 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], which yields:
φn(p)= −i (L/4πħ)1/2 exp[−ipL/(2ħ)] [ exp(+inπ/2) (sin(δn−)/δn− − exp(−inπ/2) (sin(δn+)/δn+ ], (10.18)
where δn+ ≡ (pL/ħ + nπ)/2 and δn− ≡ (pL/ħ − nπ)/2 . Note that the largest peaks in the momentum-space wave function 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 wave function arises because the position-space wave function 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 wave functions. The time is given in terms of the time it takes the ground-state wave function 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+)]. (10.19)
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.