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**Extra resources for Introductory Statistical Mechanics, Second Edition**

**Example text**

A po~sible wavefunction is cPn(x) = Bsin C;'X) where n is chosen to be an integer because we want the wavefunction to vanish at x = L. : (B sm . (~)) 8x2 L A simple model ofspins on lattice sites 73 2 2 2 • = 1i 1r n 2mL2 Bsm (~) L . The same wavefunction appears on the right-hand side multiplied by a number. The quantity Ti 21r 2n 2/2mL2 is the eigenvalue of the kinetic energy operator and Bsin(n1rx/L) is the corresponding eigenfunction. "{ample concerns the spin of a particle. Suppose a measurement is made to determine the z component of the spin of an electron.

One way of ensuringthllt this condition is satisfied is to take ¢(x) = 0 on the boundaries of the box. (Xl, X2, X3 •••• ) for the wavefunction of a many-particle system where Xl is the coordinate of the first particle and so on. The boundary conditions are then imposed on all these coordinates for a many-particle system, In quantum mechanics, physical quantities such as energy and momentulll are represented mathematically by operators which act on the wavefullction. The operator describing the energy of a system is called the Hamiltonian.

For example Xi could represent the energy of a system. 2) where Pi is the statistical probability that event i will occur. Another quantity of interest is the size of the fluctuations around the mean value. 3) can be applied to any quantity that can be represented by a 62 Probability and statistics Statistics and distributions I r-r-- 200 I I I i 2000 I r-- r- I :-- r-- 63 , r-- Ii o 88 86 29 29 Fig. he angle of deflection. histogram; it is not restricted to the counts recorded by a Geiger counter.