States, Amplitudes, and Probabilities
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.
I. Our aim here is to introduce a formal language to describe quantum ideas. First we introduce a few terms and abstract symbols.
A. A “system” is any piece of the quantum world that we wish to consider. For example, we might consider a single photon in an interferometer.
B. A “state” is a physical situation of some system.
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 “basis” is a set of distinct states that cover all of the outcomes of some measurement. For example, the photon in the interferometer would be found in one beam or the other, so the 2 states make up a basis.
There can be different possible measurements, so there can be
different possible basis sets for a quantum system.
C. Besides “basis” states, there are also “superposition” states.
1. The term superposition is meant to suggest a composite, like 2 pictures “superimposed” on one another in a double exposure.
We represent a superposition as an abstract sum:
3. 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 s2 = ½ (We give this number a special name for convenience because we will use it a lot in our examples.)
II. Next, we need rules for working and interpreting the abstract quantum symbols.
A. The “rule of superposition”
says that a superposition of 2 or more basis states is also a quantum state.
1. The means that a quantum system has more possibilities than we might expect.
For the photon in the interferometer, besides and states, we also have lots of superposition
A superposition state represents the photon divided among the beams in
some way, as happens in an interferometer. The
B. The “rule
of probability” (also called the Born rule) says that if we make a
measurement, the probability of any result is determined by the amplitude for
1. Quantum mechanics only predicts probabilities, not definite results.
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.
Suppose our photon is in the state . If we
make a measurement to find which beam the photon is in, we will get results
In the state , each beam has probability
|-s|2 = ½
C. There are 2 “update rules” that tell how the state changes when something happens to the system.
1. Update rule I says that when there is no measurement, the state changes in a definite way that maintains any superposition. If we know how to update the basis states, we can determine how to update superposition states.
2. Update rule II says that when there is a measurement, we use the results to find the new state. In this case, the state is updated randomly.
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:
These are (almost) the only rules of quantum mechanics!
III. To understand the meaning of the quantum rules, we apply them to the photon in an interferometer.
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.
The minus sign indicates reflection from the silvered side of the
In the interferometer, we keep track of the quantum state at each
The photon starts out in the upper beam, so its state is
At the first beam splitter, the state changes:
The beams recombine at the second beam splitter. We apply
We now multiply amplitudes and combine terms as we
Constructive and destructive interference take place in the
Questions to Consider:
One of the questions for the last lecture asked what happens when the
If we simply allow the 2 beams to cross without a beam splitter,
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