6.1 Specification of the state of a system
Quantum description. Any system of particles (an assembly of oscillators, a gas, a liquid, a car) is described by a wave function \(\psi(q_1,\dots,q_f)\) of \(f\) coordinates, including spin variables. The number \(f\) is the number of degrees of freedom. A quantum state is specified by a set of \(f\) quantum numbers. The description is complete: given \(\psi\) at one time, the equations of motion of quantum mechanics determine it at any other time.
Example 6.1 Describe a single spin-½ particle fixed in position.
Solution. Its state is specified by the projection \(m\) of its spin along some fixed axis (the \(z\) axis). There are two values, \(m=+\frac12\) (“up”) and \(m=-\frac12\) (“down”).
Example 6.2 Describe \(N\) spin-½ particles fixed in position.
Solution. \(N\) may be large (of order \(N_A=6\times10^{23}\)). The state is specified by the \(N\) quantum numbers \(m_1,m_2,\dots,m_N\), each \(\pm\frac12\), so there are \(2^N\) states in all.
Example 6.3 Describe a one-dimensional quantum harmonic oscillator.
Solution. With position coordinate \(x\), the states are labelled by the quantum number \(n=0,1,2,\dots\), with energy \(E_n=\big(n+\frac12\big)\hbar\omega\), where \(\omega\) is the classical angular frequency.
Example 6.4 Describe \(N\) weakly interacting one-dimensional harmonic oscillators.
Solution. The state is the set \((n_1,n_2,\dots,n_N)\), where \(n_i=0,1,2,\dots\) refers to the \(i\)-th oscillator. The total energy is \(\sum_i\big(n_i+\frac12\big)\hbar\omega\).
Example 6.5 Describe a single particle (without spin) confined in a rectangular box, \(0\le x\le L_x\), \(0\le y\le L_y\), \(0\le z\le L_z\), but otherwise free.
Solution. The wave function satisfies the Schrödinger equation \(-\dfrac{\hbar^2}{2m}\Big(\dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}+\dfrac{\partial^2}{\partial z^2}\Big)\psi=E\psi\), and must vanish on the walls. The solution is \(\psi=\sin\Big(\dfrac{\pi n_xx}{L_x}\Big)\sin\Big(\dfrac{\pi n_yy}{L_y}\Big)\sin\Big(\dfrac{\pi n_zz}{L_z}\Big)\), which vanishes at \(x=0,L_x\) (and similarly in \(y\), \(z\)) provided \(n_x,n_y,n_z\) are positive integers. Substituting gives
