187
3.3. Diffraction
satisfy
|ψ(x, t)|
2 d
3 x = 1.
(3.228)
This implies a normalization constant multiplying the wave function. As Feynman and Hibbs point out [29], there seems to be no
simple general procedure for calculating this constant. Even for
the simple case of a free-particle plane wave, we had to resort to
the device of periodic boundary conditions. Fortunately, the absolute probability is unimportant here. What is important is the
relative probability for the various point sources. This determines
the relative intensity across the beam. This becomes part of specifying the initial condition for each point source.
The fact that each point source radiates a spherical wave is equivalent to the initial momentum direction being completely unspecified. From the Heisenberg uncertainty principle, this is consistent
with the initial point (x a , t a ) being precisely specified for each
point source. The initial longitudinal momentum is precisely specified for each classical trajectory. However, the Heisenberg principle has no classical analog. We must also remember that both
the classical trajectory and the quantum mechanical wave function both apply to a single particle.
This completes the physical picture which connects the wave function to the intensity distribution of a practical system. We are now
in a position to discuss the intensity distribution for a given practical system in a more general way, through the theory of diffraction.
This forms the topic of the following sections.
3.3 Diffraction
Diffraction is the phenomenon which results from the propagation, spreading, and interference of waves. In experimental optics,
3.3. Diffraction
satisfy
|ψ(x, t)|
2 d
3 x = 1.
(3.228)
This implies a normalization constant multiplying the wave function. As Feynman and Hibbs point out [29], there seems to be no
simple general procedure for calculating this constant. Even for
the simple case of a free-particle plane wave, we had to resort to
the device of periodic boundary conditions. Fortunately, the absolute probability is unimportant here. What is important is the
relative probability for the various point sources. This determines
the relative intensity across the beam. This becomes part of specifying the initial condition for each point source.
The fact that each point source radiates a spherical wave is equivalent to the initial momentum direction being completely unspecified. From the Heisenberg uncertainty principle, this is consistent
with the initial point (x a , t a ) being precisely specified for each
point source. The initial longitudinal momentum is precisely specified for each classical trajectory. However, the Heisenberg principle has no classical analog. We must also remember that both
the classical trajectory and the quantum mechanical wave function both apply to a single particle.
This completes the physical picture which connects the wave function to the intensity distribution of a practical system. We are now
in a position to discuss the intensity distribution for a given practical system in a more general way, through the theory of diffraction.
This forms the topic of the following sections.
3.3 Diffraction
Diffraction is the phenomenon which results from the propagation, spreading, and interference of waves. In experimental optics,