Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications ofSchrödinger’s equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – no an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form ofseveral postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards ofmathematical rigour required for a strictly logical set ofaxioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8].
Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes. However, to attempt an ‘interpretation’ of the theory in order to resolve these apparent paradoxes assumes that there are natural metaphysical concepts. Suitable introductions to this topic can be found in [3, chap. 11] or [4, chap. 5].
Contents
- 14.1 Basic concepts
- 14.2 Quantum dynamics
- 14.3 Symmetry transformations
- 14.4 Quantum statistical mechanics
Concept index
Terms by section. Links lead to the brown underlined definitions in the Chinese reading text.
14.1 Basic concepts
- photon光子
- ray射线
- phase相位
- bra左矢
- ket右矢
- observable可观测量
- complete observable完备可观测量
- complementary observable互补可观测量
- compatible observable相容可观测量
14.2 Quantum dynamics
- Hamiltonian哈密顿算子
- Schrödinger picture薛定谔绘景
- Heisenberg picture海森堡绘景
- angular momentum operator角动量算子
- orbital angular momentum轨道角动量
- spin angular momentum自旋角动量
14.3 Symmetry transformations
- symmetry transformation对称变换
- antilinear transformation反线性变换
- anti-unitary operator反酉算子
- projective representation射影表示
- infinitesimal generator无穷小生成元
- Hamiltonian symmetry哈密顿对称性
- constant of the motion运动常量
- parity奇偶性
- boson玻色子
- fermion费米子