Axioms of quantum mechanics ↩

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created: 2021-12-11 08:09:12
modified: 2022-01-10 04:13:03

We can define some "axioms" in quantum mechanics, that describe the base principles of the theory. Important to note that these are not axioms in the mathematical sense that we can strictly derive everything in the theory only from these. Instead, these are just main principles of how quantum mechanics works.

Axiom 1

The state of the system is represented by a unit vector in an appropriate Hilbert space . If and are two unit vectors in with for some constant , then and represent the same physical state.

Axiom 2

To each real-valued function on the classical phase space (so classical observables) there is associated a self-adjoint operator on the quantum Hilbert space1.

Comments

  1. Since works with coordinates and momenta, these can be substituted for the position and momentum operators to become quantum mechanical.
  2. In almost all cases, the operator is unbounded. This is unsurprising, since physically relevant functions on the classical phase space (e.g. position and momentum) are unbounded functions.
  3. In most applications, it is not really necessary to define for all functions on the classical phase space, but only for certain basic functions, such as position, momentum, energy, and angular momentum.

Example for

For a particle moving in , the classical phase space is , which we think of as pairs . The quantum Hilbert space in this case is usually taken to be . In that case, if the function in Axiom 2 is the position function , then the associated operator is the position operator , given by multiplication by . We can go through similarly with the momentum operator as well.

Axiom 3

If a quantum system is in a state described by a unit vector , the probability distribution for the measurement of some observable satisfies

Where denotes the expectation value.
In particular, the expectation value for a measurement of is given by

Comments

  1. Here we have adopted the point of view that even in a quantum system, what we measure is the classical observable . In the quantum case however, no longer has a definite value, but only probabilities, which are encoded by the quantum observable and the vector .
  2. Since is assumed to be self-adjoint and every self-adjoint operator symmetric, the expectation values are real numbers.
  3. If is a unit vector and is an eigenvector of the quantum observable , then , where is the eigenvalue for . This means that if the state is an eigenvector for , then measurements for are not actually random, but rather always give the answer of . (See Born rule.)

Axiom 4

Suppose a quantum system is initially in a state and that a measurement of an observable is performed. If the result of the measurement is the number , then immediately after the measurement, the system will be in a state that satisfies

The passage from to is called the collapse of the wave function.

Comments

  1. This guarantees that if we measure and then measure again a very short time later, the result of the second measurement will agree with the result of the first measurement.

Axiom 5

The time-evolution of the wave function in a quantum system is given by the Schrödinger equation:

Where is the Hamiltonian operator of the system.

  1. The term "quantum Hilbert space" does not mean a special kind of Hilbert space, it only refers to "the Hilbert space associated with a given quantum system."
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