Lect 1 continued
We examined the two slit system
First for particles: Examine one slit open, other and both
- Defined
the density
- Particle or bullet nature of the observations
- Probability
of observing a particle at a location
Second for sound or light (waves):
- Similar shape for the intensity I(x) for either slit open.
- Double slit shows an interference pattern.
- Need to introduce amplitudes and intensities
Evaluate the sum at the same time
t and at the same location in space. However since the waves arrive from two
different slits 
.
Amplitudes:
Quantum mechanics as we will see is formulated in somewhat different ways depending on the situation. One central component in all of these formulations is the amplitude. Consider the amplitude to be the amount of a particular state present. This notion will take some experience to correctly understand.
For a classical wave 
is the amplitude. The amplitude is a convenient notion because
of the mathematical way that sine and cosine waves add together and also
because measurements and experience are often sensitive to the magnitude of the
amplitude or 
. The notation that is
used to employ this structure in a general way is 
where
is the amount of the state present. This notation will be particularly useful
when we choose our states to be normalized.
We are not prepared at this point to understand the details of
normalization but we can simply reflect on the fact that in describing 3-d
vectors we can vies components as the amount and unit vectors as the
“state”. The importance of choosing unit
vectors should be clear.
As mentioned above we will consider QM states.
This notation BRA-KET notation depicts two states of a
quantum system. Let us assume that state A has a measurable feature of value A
while state B has a measurable feature of value B. In general we will describe quantum states in
terms of amplitudes for the system to be in some set of possible states.
The classic example used to introduce these ideas is the two
slit system. Assume spherical waves
emerge from slits. Evaluate the amplitude of each wave at a point x on the
screen. Add the amplitudes. Find the magnitude squared 
(complex conjugate for
imaginary numbers).
which will be evaluated at a point on the screen x.
The phase difference between the waves is just due to difference in the path length between the two alternate routes to x.
Thus as you move up the screen the phase difference changes. The sum of the two waves (assume
) for a give angle 
and a specific time t
will be:
(Note: The details of the mathematics are not as important as the concept but are included for completeness. The point is that the sum of the waves will lead to a classic interference pattern.)
If you find magnitude squared of the sum
The cosine squared dependence shows that the amplitude will change with α. Two slit interference of waves is a standard example of how waves interfere. The key elements of importance:
- Wave intensity is the magnitude squared of the wave amplitude.
- Wave amplitude may change with position and time.
- The total wave is the sum of all the waves that arrive at the point of interest.
- Adding amplitudes and finding magnitude squared generates interference.
Finally we briefly reviewed the quantum outcomes for particles hitting the two slit system:
- Slit blocked θ expected single bump
- Both open θ wave type interference
- Open and a light source θ expected two bumps, no interference