Problem 3: Chemical potential of an Einstein solid. Consider an Einstein solid for which both N and q are much greater than 1. Think of each ocillator as a separate “particle". a) Show that the chemical potential is H = -kT In (**e) b) Discuss this result in the limits N » q and N « q, concentrating on the question of how much S increases when another particle carrying no energy is added to the system. Does the formula make intuitive sense?
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- Suppose you have a "box" in which each particle may occupy any of 10 single-particle states. For simplicity, assume that each of these states has energy zero. What is the partition function of this system if the box contains only one particle?PV 1/V- (1) (ii) 1/V (iv) (ii) Boyle's law may be best expressed by O the top left curve any of the curves shown here either of the curves in the bottom row O the top right curveProblem 1: (a) A non-relativistic, free particle of mass m is bouncing back and forth between two perfectly reflecting walls separated by a distance L. Imagine that the two oppositely directed matter waves associated with this particle interfere to create a standing wave with a node at each of the walls. Find the kinetic energies of the ground state (first harmonic, n = 1) and first excited state (second harmonic, n = 2). Find the formula for the kinetic energy of the n-th harmonic. (b) If an electron and a proton have the same non-relativistic kinetic energy, which particle has the larger de Broglie wavelength? (c) Find the de Broglie wavelength of an electron that is accelerated from rest through a small potential difference V. (d) If a free electron has a de Broglie wavelength equal to the diameter of Bohr's model of the hydrogen atom (twice the Bohr radius), how does its kinetic energy compare to the ground-state energy of an electron bound to a Bohr model hydrogen atom?
- Problem 3.36. Consider an Einstein solid for which both N and q are much greater than 1. Think of each oscillator as a separate "particle." (a) Show that the chemical potential is N+ - kT ln N (b) Discuss this result in the limits N > q and N « q, concentrating on the question of how much S increases when another particle carrying no energy is added to the system. Does the formula make intuitive sense?Problem 7. 1. Calculate the energy of a particle subject to the potential V(x) = Vo + câ?/2 if the particle is in the third excited state. 2. Calculate the energy eigenvalues for a particle moving in the potential V(x) = câ2/2+ bx. %3!Please don´t answer by partition equation!
- Consider the three-dimensional harmonic oscillator, for which the potential is V ( r ) = 1/2 m ω2 r2 (a) Show that the separation of variables in Cartesian coordinates turns this into three one-dimensional oscillators, and exploit your knowledge of the latter to determine the allowed energies. Answer: En = ( n + 3/2 ) ħ ω (b) Determine the degeneracy d ( n ) of EnIn discussing the velocity distribution of molecules of an ideal gas, a functionF(x, y, z) = f(x)f(y)f(z) is needed such that d(ln F) = 0 when φ = x2 + y2 + z2 =const. Then by the Lagrange multiplier method d(ln F + λφ) = 0. Use this to show that F(x, y, z) = Ae−(λ/2)(x2+y2+z2).Problem 1: Estimate the probability that a hydrogen atom at room temperature is in one of its first excited states (relative to the probability of being in the ground state). Don't forget to take degeneracy into account. Then repeat the calculation for a hydrogen atom in the atmosphere of the star y UMa, whose surface temperature is approximately 9500K.
- Problem 1. Using the WKB approximation, calculate the energy eigenvalues En of a quantum- mechanical particle with mass m and potential energy V (x) = V₁ (x/x)*, where V > 0, Express En as a function of n; determine the dimensionless numeric coefficient that emerges in this expression.Question 2: The Rigid Rotor Consider two particles of mass m attached to the ends of a massless rigid rod with length a. The system is free to rotate in 3 dimensions about the center, but the center point is held fixed. (a) Show that the allowed energies for this rigid rotor are: n?n(n+1) En = where n = 0, 1, 2... та? (b) What are the normalized eigenfunctions for this system? (c) A nitrogen molecule (N2) can be described as a rigid rotor. If the distance between N atoms is 1.1 Å, then what wavelength of photon must be absorbed to produce a transition from the n = 1 to n = 4 state?Consider a 10000 distinguishable particles at room temperature, 298 K. Suppose that each particle has 2 energy levels, 0.01 eV and 0.02 eV. Find the number of the particles in each energy level. ( Boltzmann's constant is 1.3807 × 10-23 J K-1.)