3.3 Dynamics of a two-state system

对照

Recall that the time dependence of states in quantum mechanics is given by the Schrödinger equation

$$\frac{d}{dt}|\psi (t)⟩=-iH|\psi (t)⟩$$

where $H$ is a particular self-adjoint linear operator on $\mathcal{H}$ , the Hamiltonian operator. Considering the case of $H$ time-independent, the most general such operator $H$ on ${\mathbf{C}}^2$ will be given by

$$H=h_{0}1+h_1{\sigma }_1+h_2{\sigma }_2+h_3{\sigma }_3$$

for four real parameters $h_0,h_1,h_2,h_3$ . The solution to the Schrödinger equation is then given by exponentiation:

$$|\psi (t)⟩=U(t)|\psi (0)⟩$$

where

$$U(t)=e^{-itH}$$

The $h_{0}1$ term in $H$ contributes an overall phase factor $e^{-i{h}_{0}t}$ , with the remaining factor of $U(t)$ an element of the group $SU(2)$ rather than the larger group $U(2)$ of all 2 by 2 unitaries.

Using our equation 3.3, valid for a unit vector $\mathbf{v}$, our $U(t)$ is given by taking $\mathbf{h}=(h_1,h_2,h_3)$, $\mathbf{v}=\frac{\mathbf{h}}{|\mathbf{h}|}$ and $\theta =-t|\mathbf{h}|$, so we find

$$\begin{array}{rl}U(t)& =e^{-i{h}_{0}t}\left(\cos (-t|\mathbf{h}|)1+i\sin (-t|\mathbf{h}|)\frac{h_1{\sigma }_1+h_2{\sigma }_2+h_3{\sigma }_3}{|\mathbf{h}|}\right)\\ & =e^{-i{h}_{0}t}\left(\cos (t|\mathbf{h}|)1-i\sin (t|\mathbf{h}|)\frac{h_1{\sigma }_1+h_2{\sigma }_2+h_3{\sigma }_3}{|\mathbf{h}|}\right)\\ & =e^{-i{h}_{0}t}\left(\begin{array}{cc}\cos (t|\mathbf{h}|)-i\frac{h_3}{|\mathbf{h}|}\sin (t|\mathbf{h}|)& -i\sin (t|\mathbf{h}|)\frac{h_1-i{h}_2}{|\mathbf{h}|}\\ -i\sin (t|\mathbf{h}|)\frac{h_1+i{h}_2}{|\mathbf{h}|}& \cos (t|\mathbf{h}|)+i\frac{h_3}{|\mathbf{h}|}\sin (t|\mathbf{h}|)\end{array}\right)\end{array}$$

In the special case $\mathbf{h}=(0,0,h_3)$ we have

$$U(t)=\left(\begin{array}{cc}{e}^{-it(h_0+h_3)}& 0\\ 0& e^{-it(h_0-h_3)}\end{array}\right)$$

so if our initial state is

$$|\psi (0)⟩=\alpha |+\frac{1}{2}⟩+\beta |-\frac{1}{2}⟩$$

for $\alpha ,\beta \in \mathbf{C}$ , at later times the state will be

$$|\psi (t)⟩=\alpha e^{-it(h_0+h_3)}|+\frac{1}{2}⟩+\beta e^{-it(h_0-h_3)}|-\frac{1}{2}⟩$$

In this special case, the eigenvalues of the Hamiltonian are $h_0\pm h_3$

In the physical realization of this system by a spin $\frac{1}{2}$ particle (ignoring its spatial motion), the Hamiltonian is given by

$$H=\frac{ge}{4mc}(B_1{\sigma }_1+B_2{\sigma }_2+B_3{\sigma }_3)\text{\hspace{1em}(3.6)}$$

where the $B_j$ are the components of the magnetic field, and the physical constants are the gyromagnetic ratio $(g)$, the electric charge $(e)$, the mass $(m)$ and the speed of light $(c)$. By computing $U(t)$ above, we have solved the problem of finding the time evolution of such a system, setting $h_j=\frac{ge}{4mc}{B}_j$. For the special case of a magnetic field in the 3-direction $(B_1=B_2=0)$, we see that the two different states with well-defined energy ($|+\frac{1}{2}⟩$ and $|-\frac{1}{2}⟩$, recall that the energy is the eigenvalue of the Hamiltonian) will have an energy difference between them of

$$2h_3=\frac{ge}{2mc}{B}_3$$

This is known as the Zeeman effect and is readily visible in the spectra of atoms subjected to a magnetic field. We will consider this example in more detail in chapter 7, seeing how the group of rotations of ${\mathbf{R}}^3$ enters into the story. Much later, in chapter 45, we will derive the Hamiltonian 3.6 from general principles of how electromagnetic fields couple to spin $\frac{1}{2}$ particles.

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上一节：3.2 Commutation relations for Pauli matrices · 下一节：3.4 For further reading

附录：约定与记号

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正文：英文 · 原著转录

核对状态：AI 辅助转录核对，未作人工审阅

原书 PDF · 印刷页 32、33

来源版本：2025-10-20

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