FOUNDATIONS OF QUANTUM MECHANICS

IV. 5. THE UNCERTAINTY RELATIONS 101
2 a
|ψ ss (q)| 2
Figure IV. 6: The probability distribution in position for a slit of width 2 a
The Fourier transform of ψ ss is
˜ψ ss (p) =
√ a
π
sin(ap/)
. (IV. 44)
a p /
The square of this wave function, | ˜ψ ss (p)| 2 , has the same form as the diffraction pattern for the slit
which is formed on a photographic plate placed far away, see figure IV. 7.
2π/a
| ˜ψ ss (p)| 2
Figure IV. 7: The diffraction pattern for a small slit of width 2 a
For the standard deviation of position and momentum in the state ψ ss we find
(∆ ψss Q) 2 =
∫
R
q 2 |ψ ss (q)| 2 dq = 1
2 a
∫ +a
−a
q 2 dq = 1 3 a2 (IV. 45)