Essential University Physics (3rd Edition)
3rd Edition
ISBN: 9780134202709
Author: Richard Wolfson
Publisher: PEARSON
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Chapter 36, Problem 47P
(a)
To determine
The energy of the eight electron in harmonic oscillator potential and in its lowest-energy state.
(b)
To determine
The lowest energy of the eight spin-1 particles in harmonic oscillator potential
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(4) Electronic energy level of a hydrogen atom is given by
R
E
; n =
n2
1, 2, 3,...
and R = 13.6 eV. Each energy level has degeneracy 2n² (degeneracy is the number of equivalent
configurations associated with the energy level).
(a) Calculate the partition function Z for a hydrogen atom at a constant temperature.
(b) Let us consider that the energy level of a hydrogen atom is approximated by a two level
system, n = 1,2. Estimate the mean energy at 300 K.
(6) (a) Show that the speed of an electron in the nth Bohr orbit of hydrogen is ac/n where a is the
fine structure constant.
(b) What would be the speed in a hydrogenlike atom with a nuclear charge of Ze?
The degeneracy pressure of the electrons can stabilize the collapse of a star due to gravity by equating the gravitational inward pressure with the outward electron gas degeneracy pressure. These cold stars called white dwarfs have small radii compared to their original size and this radius decreases as the original mass of the star increases. As the mass of a star increases, the electron energy increases to a point in which their energy has to be treated relativistically.
(a) Evaluate the degeneracy pressure for ultra-relativistic electrons (problem 2, above).
(b) As the pressure increases, the reaction e − + p → n + ν takes place. The neutrinos (ν) escape as matter is transparent to them, electrons and protons convert to neutrons until we are left with a neutron star. Using your expression for the degeneracy pressure obtained in (a) above, equate the gravitational pressureto the neutron degeneracy pressure (replace the electron’s mass in your degeneracy pressure expression from (a)…
Chapter 36 Solutions
Essential University Physics (3rd Edition)
Ch. 36.1 - Prob. 36.1GICh. 36.2 - Prob. 36.2GICh. 36.3 - Prob. 36.3GICh. 36.4 - Prob. 36.4GICh. 36.5 - Prob. 36.5GICh. 36 - Prob. 1FTDCh. 36 - Prob. 2FTDCh. 36 - Prob. 3FTDCh. 36 - Prob. 4FTDCh. 36 - Prob. 5FTD
Ch. 36 - Prob. 6FTDCh. 36 - Prob. 7FTDCh. 36 - Prob. 8FTDCh. 36 - Prob. 9FTDCh. 36 - Prob. 10FTDCh. 36 - Prob. 11FTDCh. 36 - Prob. 12FTDCh. 36 - What distinguishes a Bose-Einstein condensate from...Ch. 36 - Prob. 14ECh. 36 - Prob. 15ECh. 36 - Prob. 16ECh. 36 - Prob. 17ECh. 36 - Prob. 18ECh. 36 - Prob. 19ECh. 36 - Prob. 20ECh. 36 - Prob. 21ECh. 36 - Prob. 22ECh. 36 - Prob. 23ECh. 36 - Prob. 24ECh. 36 - Prob. 25ECh. 36 - Prob. 26ECh. 36 - Prob. 27ECh. 36 - Prob. 28ECh. 36 - Prob. 29ECh. 36 - Prob. 30ECh. 36 - Prob. 31ECh. 36 - Prob. 32ECh. 36 - Prob. 33ECh. 36 - Prob. 34PCh. 36 - Prob. 35PCh. 36 - Prob. 36PCh. 36 - Prob. 37PCh. 36 - Prob. 38PCh. 36 - Prob. 39PCh. 36 - Prob. 40PCh. 36 - Prob. 41PCh. 36 - Prob. 42PCh. 36 - Prob. 43PCh. 36 - Prob. 44PCh. 36 - Prob. 45PCh. 36 - Prob. 46PCh. 36 - Prob. 47PCh. 36 - Prob. 48PCh. 36 - Prob. 49PCh. 36 - Prob. 50PCh. 36 - Prob. 51PCh. 36 - Prob. 52PCh. 36 - Prob. 53PCh. 36 - Prob. 54PCh. 36 - Prob. 55PCh. 36 - Prob. 56PCh. 36 - Prob. 57PCh. 36 - Prob. 58PCh. 36 - Prob. 59PCh. 36 - Prob. 60PCh. 36 - Prob. 61PCh. 36 - Prob. 62PCh. 36 - Prob. 63PCh. 36 - Prob. 64PCh. 36 - Prob. 65PCh. 36 - Prob. 66PCh. 36 - Prob. 67PCh. 36 - Prob. 68PCh. 36 - Prob. 69PCh. 36 - Prob. 70PCh. 36 - Prob. 71PCh. 36 - Prob. 72PCh. 36 - Prob. 73PCh. 36 - Prob. 74PCh. 36 - Prob. 75PCh. 36 - Prob. 76PPCh. 36 - Prob. 77PPCh. 36 - Prob. 78PPCh. 36 - Prob. 79PP
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- Comparisons of the energy differences between two states with that predicted by theory can test the theoretical model. For the case of the muon g-value, there's a small discrepancy between it and the Standard Model of particle physics. We can see atomic transitions in cesium in atomic clocks. We can measure the frequency by applying pulses at frequencies to observe the resonance (see the provided equation, Rabi's formula). Consider a spin-1/2 particle that precesses in a magnetic field in the z direction. The probability of the particle being spin up or spin down along z doesn’t vary with time. The states |+z> and |—z> are stationary states of the Hamiltonian H-hat = ω0 S-hatz. If we alter the Hamiltonian by applying in addition an oscillating magnetic field transverse to the z axis, we can induce transitions between these two states by properly adjusting the frequency of this transverse field. The energy difference E+—E_ = h-bar ω0 can then be measured with high accuracy. This…arrow_forward(a) Show that the speed of an electron in the nth Bohr orbit of hydrogen is ac/n, where a is the fine structure constant, equal to e/4neghc. (b) What would be the speed in a hydrogen like atom with a nuclear charge of Ze?arrow_forwardWhat is the probability that an electron in the base state of the hydrogen atom will be found inside the nucleus? (let the radius of the nucleus be r).arrow_forward
- Please asaparrow_forwardAssume that the nucleus of an atom can be regarded as a three-dimensional box of width 2:10-¹4 m. If a proton moves as a particle in this box, find (a) the ground-state energy of proton in MeV and (b) the energies of the first excited state. (c) What are the degenerates of these states? Constants: h = 6.626-10-34 [J-s], m = 1.673-10-27 [kg] and ħ=h/2π.arrow_forwardA spin half particle is prepared in the spin state |S) = 2|+) + (1 + V3i)l-). where |S) = |S - n = th). What is the component în of spin of the particle which is known to be th?arrow_forward
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- (a) Calculate: (i) the energy spacing AE between the ground state and the first excited state of the hydrogen atom; (ii) and the ratio AE/E between the spacing and the ground state energy. (b) Consider now a macroscopic system: a simple pendulum which consists of a 5 g mass attached to a 2 m long, massless and inextensible string. Calculate (i) the total energy E1 of the pendulum when the string makes an angle of 60° with the vertical; (ii) the frequeney of the pendulum's small oscillations and the energy AE of one quantum; and (iii) the ratio AE/E1. (c) Examine the sizes of the ratio AE/E1 calculated in parts (a) and (b) and comment on the importance of the quantum effects for the hydrogen atom and the pendulum.arrow_forwardFind the binding energy of the hydrogen electron for states with the following principal quantum numbers. (Note the binding energy is just the absolute value of the Energy level, the energy required to remove the electron from that state to infinity.) a) For n = 3. b) For n = 5. c) For n = 7.arrow_forwardWhat is the maximum kinetic energy of an electron such that a collision between the electron and a stationary hydrogen atom in its ground state is definitely elastic?arrow_forward
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