References: Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Sections 4.1 – 4.2; Exercise 4.2.1.
Although we’ve covered the basics of nonrelativistic quantum mechanics before, the approach taken by Shankar in his Chapter 4 provides a new way of looking at it, so it’s worth a summary.
Quantum mechanics is based on four postulates, the first three of which describe the quantum state at a fixed instant in time, and the fourth which describes its time evolution via the Schrödinger equation. We’ll summarize the first three postulates here, and compare each with its classical analogue.
First, in classical mechanics, the path of a particle is, in the Hamiltonian formalism, described by specifying its position and momentum as functions of time. Both the position and momentum are specified precisely at all times. In quantum mechanics, the state of a particle is specified by a vector (ket) in a Hilbert space.
Second, in classical mechanics, any dynamical variable is a function of the two phase-space coordinates and : . In quantum mechanics, the spatial coordinate is replaced by the Hermitian operator and the momentum is replaced by the differential operator which we discussed earlier. The matrix elements of and in position space are
The classical dynamical variable becomes a Hermitian operator , where and in are replaced by their corresponding operators and .
The third postulate states how measurements work in quantum mechanics. In classical mechanics, it is assumed that (in principle) any dynamical variable may be measured with arbitrary precision without changing the state of the particle. In quantum mechanics, if we wish to measure the value of a variable represented by the operator , we must determine the eigenvalues and corresponding eigenvectors of , then express the state as a linear combination of the . Then the best we can do is to state that the particular eigenvalue will be measured with probability . After the measurement, the state ‘collapses’ to become the state . The only possible outcomes of a measurement of are its eigenvalues; no intermediate values are possible.
To illustrate these postulates, suppose we have the following three operators on a complex 3-d Hilbert space (essentially these are the spin-1 operators without the )
Since is diagonal, its eigenvalues can be read off from the diagonal elements as , so these are the possible values of that could be obtained in a measurement. Also because is diagonal, its eigenvectors are
Suppose we start with the state in which , and we want to measure in this state. To find the expectation values and in this state, we calculate
To get we first find the operator
Now we have
The uncertainty, or variance, is
To find the possible values of and their probabilities, we need to find the eigenvalues and eigenvectors of , which we can do in the basis, since this basis is given by the three vectors in 6. The eigenvalues are found in the usual way from the determinant:
The eigenvectors can be found in the usual way, by solving
where the ket takes on the three possible values of successively. We let
For we have
Only two of these three equations are independent, so we can set and solve for and to get
Normalizing the eigenvector gives
The other two eigenvectors can be found the same way, with the result
Note that these eigenvectors are orthonormal.
Now that we have the eigenvectors of we can answer the following question. If we start with the state and measure , what are the possible outcomes and the probability of each?
First, we need to express in terms of the eigenvectors of which we can do by solving three simultaneous linear equations, and we find
(You can verify this by direct substitution.) Thus all 3 possible values of can result from a measurement, and the probability of each is
Now suppose we start with the state, written in the basis:
We take a measurement of and obtain . The operator is given by squaring 5:
This has a degenerate eigenvalue , so the most we can say about the state after the measurement is that it is projected onto the subspace . That is, the state after the measurement is given by
We can normalize this state to get
Thus if we measure immediately after the measurement of above, we get with probability and with probability .
Finally, suppose we have a state with the probabilities of measurements of given as , and . Since these probabilities are given by for each of the three possible values of , and the vectors are orthonormal, the most general form for is
where the are real numbers. For example
While the presence of a phase factor in a solitary state doesn’t affect the physics of that state, if we have a sum of states, each with its own (different) phase factor, we can’t ignore these phase factors. For example, if we measure in this state and want the probability that , we have, using 30
The last line will have a different result for different values of the phase factors and , so they can’t be ignored.