Scope: The
simple photon interferometer example from Lecture Eight is our doorway into
the formal math of quantum theory. By learning a few symbols and a handful of
rules for manipulating and interpreting them, we can describe the states of
quantum particles, show how these states change over time, and predict the
results of measurements. One key idea is the principle of superposition,
which tells us how to combine 2 quantum states to form a new one. Another
concept is the probability amplitude, used to calculate the likelihood of the
various outcomes of an experiment.
A. A “ B.
A “ 1.
We represent a state by a “ket,” like so: . What we
put inside the ket is just a convenient label for the state 2.
A “ 3.
There can be different possible measurements, so there can be
different possible basis sets for a quantum system. C. Besides “ 1.
The term superposition is meant to suggest a composite, like 2
pictures “superimposed” on one another in a double exposure. 2.
We represent a superposition as an abstract sum: are
called “amplitudes.”b3.
In full quantum mechanics, these amplitudes might include imaginary numbers . We can
omit this complication, but we will use both positive and negative
amplitudes. 4.
We define the numbers = 0.7071 ... , for
which s
A. The “ 1.
The means that a quantum system has more possibilities than we might
expect. 2.
For the photon in the interferometer, besides and states, we also have lots of superposition
states 3.
A superposition state represents the photon divided among the beams in
some way, as happens in an interferometer. The B. The probability = |ampIitude| ^{2}. 1.
Quantum mechanics only predicts probabilities, not definite results. 2.
What is probability? For any event, its probability P is a number between
0 and 1. The value P = 0 means the event is impossible, and P = 1 means that
it is certain. An intermediate value like P = 0.37 means that, if we tried
the same experiment many times, the event would happen about 37% of the time.
Probabilities predict statistics. 3.
Both positive and negative amplitudes give positive probabilities. 4.
Suppose our photon is in the state . If we
make a measurement to find which beam the photon is in, we will get results
with probabilities: 5.
In the state , each beam has probability |s| C. There are 2 “update
rules” that tell how the state changes when something happens to the system. 1.
Update 2.
Update 3.
An example for update rule II: The photon is in the , and we
use photon detectors to determine which beam it is in. Then: D.
These are (almost) the only rules of quantum mechanics!
A.
At a beam splitter, the basis states change in this way:
1.
This is an example of update rule I, since no measurement is made. 2.
The minus sign indicates reflection from the silvered side of the
mirror. B.
In the interferometer, we keep track of the quantum state at each 1.
The photon starts out in the upper beam, so its state is 2.
At the first beam splitter, the state changes: 3.
The beams recombine at the second beam splitter. We apply 4.
We now multiply amplitudes and combine terms as we 5.
Constructive and destructive interference take place in the
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