Exercise 4.2.2 Consider a particle in initial spin up. Apply S, to it and determine the probability that the resulting state is still spin up.
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- Problem 2.8 A particle of mass m in the infinite square well (of width a) starts out in the left half of the well, and is (at t = 0) equally likely to be found at any point in that region. (a) What is its initial wave function, (x, 0)? (Assume it is real. Don't forget to normalize it.) (b) What is the probability that a measurement of the energy would yield the value л²ħ²/2ma²?5.4 If the accelerating potential between the cathode and anode of Thomson's elme apparatus is 182.2 V, what uniform velocity v will the electrons acquire before entering the coexisting E and B fields? Assume accuracy to three significant figures and derive the appropriate equation.6.2 Let the "uniform" ensemble of energy E be defined as the ensemble of all systems of the given type with energy less than E. The equivalence between (6.29) and (6.27) means that we should obtain the same thermodynamic functions from the "uniform" ensemble of energy E as from the microcanonical ensemble of energy E. In particular, the internal energy is E in both ensembles. Explain why this seemingly paradoxical result is true.
- Exercise 1.13. Show that the hat function basis {i}"o of Vh is almost orthogonal. How can we see that it is almost orthogonal by looking at the non-zero elements of the mass matrix? What can we say about the mass matrix if we had a fully orthogo- nal basis?4Problem 4.25 If electron, radius [4.138] 4πεmc2 What would be the velocity of a point on the "equator" in m /s if it were a classical solid sphere with a given angular momentum of (1/2) h? (The classical electron radius, re, is obtained by assuming that the mass of the electron can be attributed to the energy stored in its electric field with the help of Einstein's formula E = mc2). Does this model make sense? (In fact, the experimentally determined radius of the electron is much smaller than re, making this problem worse).
- Exercise 6.4 Consider an anisotropic three-dimensional harmonic oscillator potential acy = { m (w² x ² + w} y² + w? 2²). V (x, y, z) = = m(o² x² + @z. (a) Evaluate the energy levels in terms of wx, @y, and (b) Calculate [Ĥ, Î₂]. Do you expect the wave functions to be eigenfunctions of 1²? (c) Find the three lowest levels for the case @x = @y= = 2002/3, and determine the degener- of each level.Problem 4.45 What is the probability that an electron in the ground state of hydro- gen will be found inside the nucleus? (a) First calculate the exact answer, assuming the wave function (Equation 4.80) is correct all the way down to r = 0. Let b be the radius of the nucleus. (b) Expand your result as a power series in the small number € = 2b/a, and show that the lowest-order term is the cubic: P≈ (4/3)(b/a)³. This should be a suitable approximation, provided that bplease answer quickly1.2 It is given that the primitive basis vectors of a lattice are: a = 3%, b= 3ý and c=&+ŷ +i) What is the Bravais lattice?Problem 4.38. Two magnetic systems in thermal contact Consider two isolated systems of noninteracting spins with NA energies are EA = -2µB and Eg = -2µB. = 4 and NB = 16. Their initial (a) What is the total number of microstates available to the composite system? (b) If the two systems are now allowed to exchange energy with one another, what is the probability that system A has energy EA? (c) What is the mean value of EA and its relative fluctuations? Calculate the analogous quantities for system B.