1.2 Basic principles of quantum mechanics

对照

We’ll divide the conventional list of basic principles of quantum mechanics into two parts, with the first covering the fundamental mathematics structures.

1.2.1 Fundamental axioms of quantum mechanics

In classical physics, the state of a system is given by a point in a “phase space”, which can be thought of equivalently as the space of solutions of an equation of motion, or as (parametrizing solutions by initial value data) the space of coordinates and momenta. Observable quantities are just functions on this space (e.g., functions of the coordinates and momenta). There is one distinguished observable, the energy or Hamiltonian, and it determines how states evolve in time through Hamilton’s equations.

The basic structure of quantum mechanics is quite different, with the formalism built on the following simple axioms:

Axiom (States). The state of a quantum mechanical system is given by a nonzero vector in a complex vector space $\mathcal{H}$ with Hermitian inner product $⟨\cdot ,\cdot ⟩$

We’ll review in chapter 4 some linear algebra, including the properties of inner products on complex vector spaces. $\mathcal{H}$ may be finite or infinite dimensional, with further restrictions required in the infinite dimensional case (e.g., we may want to require $\mathcal{H}$ to be a Hilbert space). Note two very important diferences with classical mechanical states:

• The state space is always linear: a linear combination of states is also a state.

• The state space is a complex vector space: these linear combinations can and do crucially involve complex numbers, in an inescapable way. In the classical case only real numbers appear, with complex numbers used only as an inessential calculational tool.

We will sometimes use the notation introduced by Dirac for vectors in the state space $\mathcal{H}$: such a vector with a label $\psi$ is denoted

$$|\psi ⟩$$

Axiom (Quantum observables). The observables of a quantum mechanical system are given by self-adjoint linear operators on $\mathcal{H}$.

We’ll review the definition of self-adjointness for $\mathcal{H}$ finite dimensional in chapter 4. For $\mathcal{H}$ infinite dimensional, the definition becomes much more subtle, and we will not enter into the analysis needed.

Axiom (Dynamics). There is a distinguished quantum observable, the Hamiltonian $H$. Time evolution of states $|\psi (t)⟩\in \mathcal{H}$ is given by the Schrödinger equation

$$i\hslash \frac{d}{dt}|\psi (t)⟩=H|\psi (t)⟩\text{\hspace{1em}(1.1)}$$

The operator $H$ has eigenvalues that are bounded below.

The Hamiltonian observable $H$ will have a physical interpretation in terms of energy, with the boundedness condition necessary in order to assure the existence of a stable lowest energy state.

$\hslash$ is a dimensional constant, called Planck’s constant, the value of which depends on what units one uses for time and for energy. It has the dimensions [energy] · [time] and its experimental values are

$1.054571726(47)\times 10^{-34}\mathrm{Joule}\cdot \mathrm{seconds}=6.58211928(15)\times 10^{-16}\mathrm{eV}\cdot \mathrm{seconds}$

(eV is the unit of “electron-Volt” , the energy acquired by an electron moving through a one-Volt electric potential). The most natural units to use for quantum mechanical problems would be energy and time units chosen so that $\hslash =1$. For instance one could use seconds for time and measure energies in the very small units of $6.6\times 10^{-16}\mathrm{eV}$ , or use eV for energies, and then the very small units of $6.6\times 10^{-16}$ seconds for time. Schrödinger’s equation implies that if one is looking at a system where the typical energy scale is an $\mathrm{eV}$ , one’s state-vector will be changing on the very short time scale of $6.6\times 10^{-16}$ seconds. When we do computations, usually we will set $\hslash =1$ , implicitly going to a unit system natural for quantum mechanics. After calculating a final result, appropriate factors of $\hslash$ can be inserted to get answers in more conventional unit systems.

It is sometimes convenient however to carry along factors of $\hslash$, since this can help make clear which terms correspond to classical physics behavior, and which ones are purely quantum mechanical in nature. Typically classical physics comes about in the limit where

$\frac{(\text{energy scale})(\text{time scale})}{\hslash }$

is large. This is true for the energy and time scales encountered in everyday life, but it can also always be achieved by taking $\hslash \to 0$ , and this is what will often be referred to as the “classical limit”. One should keep in mind though that the manner in which classical behavior emerges out of quantum theory in such a limit can be a very complicated phenomenon.

1.2.2 Principles of measurement theory

The above axioms characterize the mathematical structure of a quantum theory, but they don’t address the “measurement problem”. This is the question of how to apply this structure to a physical system interacting with some sort of macroscopic, human-scale experimental apparatus that “measures” what is going on. This is a highly thorny issue, requiring in principle the study of two interacting quantum systems (the one being measured, and the measurement apparatus) in an overall state that is not just the product of the two states, but is highly “entangled” (for the meaning of this term, see chapter 9). Since a macroscopic apparatus will involve something like $10^{23}$ degrees of freedom, this question is extremely hard to analyze purely within the quantum mechanical framework (requiring for instance the solution of a Schrödinger equation in $10^{23}$ variables).

Instead of trying to resolve in general this problem of how macroscopic classical physics behavior emerges in a measurement process, one can adopt the following two principles as providing a phenomenological description of what will happen, and these allow one to make precise statistical predictions using quantum theory:

Principle (Observables). States for which the value of an observable can be characterized by a well-defined number are the states that are eigenvectors for the corresponding self-adjoint operator. The value of the observable in such a state will be a real number, the eigenvalue of the operator.

This principle identifies the states we have some hope of sensibly associating a label to (the eigenvalue), a label which in some contexts corresponds to an observable quantity characterizing states in classical mechanics. The observables with important physical significance (for instance the energy, momentum, angular momentum, or charge) will turn out to correspond to some group action on the physical system.

Principle (The Born rule). Given an observable $O$ and two unit-norm states $|{\psi }_1⟩$ and $|{\psi }_2⟩$ that are eigenvectors of $O$ with distinct eigenvalues ${\lambda }_1$ and ${\lambda }_2$

$$O|{\psi }_1⟩={\lambda }_1|{\psi }_1⟩,O|{\psi }_2⟩={\lambda }_2|{\psi }_2⟩$$

the complex linear combination state

$$c_1|{\psi }_1⟩+c_2|{\psi }_2⟩$$

will not have a well-defined value for the observable $O$ . If one attempts to measure this observable, one will get either ${\lambda }_1$ or ${\lambda }_2$ , with probabilities

$$\frac{|c_1^2|}{|c_1^2|+|c_2^2|}$$

and

$$\frac{|c_2^2|}{|c_1^2|+|c_2^2|}$$

respectively.

The Born rule is sometimes raised to the level of an axiom of the theory, but it is plausible to expect that, given a full understanding of how measurements work, it can be derived from the more fundamental axioms of the previous section. Such an understanding though of how classical behavior emerges in experiments is a very challenging topic, with the notion of “decoherence” playing an important role. See the end of this chapter for some references that discuss these issues in detail.

Note that the state $c|\psi ⟩$ will have the same eigenvalues and probabilities as the state $|\psi ⟩$, for any complex number $c$. It is conventional to work with states of norm fixed to the value 1, which fixes the amplitude of $c,$ leaving a remaining ambiguity which is a phase $e^{i\theta }$ . By the above principles this phase will not contribute to the calculated probabilities of measurements. We will however not take the point of view that this phase information can just be ignored. It plays an important role in the mathematical structure, and the relative phase of two different states certainly does affect measurement probabilities.

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上一节：1.1 Introduction · 下一节：1.3 Unitary group representations

附录：约定与记号

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