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Chapter 8 Problems

Chapter 8 Problems - Problems Numericalproblems...

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Problems Numerical problems 8.1   The Planck distribution gives the energy in the wavelength range d λ  at the wavelength  λ . Calculate the energy  density in the range 650 nm to 655 nm inside a cavity of volume 100 cm 3  when its temperature is (a) 25 ° C, (b)  3000 ° C. 8.2    For a  black body , the  temperature  and the wavelength of emission maximum,  λ max ,are related by Wien’s law,  λ max T  =  1 / 5 c 2 ,where  c 2  =  hc / k  (see  Problem   8.10 ). Values of  λ max  from a small pinhole in an electrically heated  container were determined at a series of  temperatures ,and the results are given below. Deduce a value for Planck’s  constant. 8.3   The Einstein frequency is often expressed in terms of an equivalent temperature  θ E , where  θ E  =  h ν / k . Confirm  that  θ E  has the dimensions of temperature,and express the criterion for the validity of the high-temperature form of  the Einstein equation in terms of it. Evaluate  θ E  for (a) diamond, for which  ν  = 46.5 THz and (b) for copper, for which  ν  = 7.15 THz. What fraction of the Dulong and Petit value of the heat capacity does each substance reach at 25 ° C? 8.4    The ground-state  wavefunction  for a particle confined to a one-dimensional box of length  L  is Suppose the box is 10.0 nm long. Calculate the probability that the particle is (a) between  x  = 4.95 nm and 5.05 nm,  (b) between  x  = 1.95 nm and 2.05 nm, (c) between  x  = 9.90 nm and 10.00 nm, (d) in the right half of the box, (e) in  the central third of the box. 8.5   The ground-state wavefunction of a hydrogen atom is where  a 0  = 53 pm (the Bohr radius). (a) Calculate the probability that the electron will be found somewhere within a  small sphere of radius 1.0 pm centred on the nucleus. (b) Now suppose that the same sphere is located at  r  =  a 0 What is the probability that the electron is inside it? 8.6   The normalized wavefunctions for a particle confined to move on a circle are  ψ ( φ ) = (1/2 π ) 1/2 e i m φ ,where  m  = 0,  ± 1,  ± 2,  ± 3,...and 0      φ  2 π . Determine  φ . 8.7   A particle is in a state described by the wavefunction  ψ ( x ) = (2 a / π ) 1/4 e ax 2, where  a  is a constant and  −∞    x    Verify that the value of the product  p x  is consistent with the predictions from the uncertainty principle.
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