For sinusoidal perturbation, H'(F,1)=VF)cos(x), show that the transition probability is given by P,(1) = (Vf /h°)(sin*[(@, – @)t / 2]/(@, – @)°) where V = (w.V(#)w)
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![For sinusoidal perturbation, H'(F,1)=VF)cos(x), show that the transition probability
is given by
P,(1) = (Vf /h°)(sin*[(@, – @)t / 2]/(@, – @)°)
where V = (w.V(#)w)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ce2a06e-a73c-4e84-a56e-0b767511adc4%2F2baa2e31-9585-4d0f-b680-a93ead6c02da%2F6c6ulzh.jpeg&w=3840&q=75)
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- Show that at high enough temperatures (where KBT » ħw) the partition function of a simple quantum mechanical harmonic oscillator is approximately Z≈ (Bħw)-¹ Then use the partition function to calculate the high temperature expressions for the internal energy U, the heat capacity Cy, the Helmholtz function F and the entropy S.(1) A single particla quantum mechanical oscillator has energy levels (n + 1/2) hw, where n = 0, 1, 2, .. and w is the natural frequency of the oscillator. This oscillator is in thermal equi- librium with a reservoir at temperature T. (a) Find the ratio of probability of the oscillator being in the first excited state (n = 1) to the probability of being in the ground state. (b) Assuming that only the two states in Part la are occupied, find the average energy as a function of T. (c) Calculate the heat capacity at a constant volume. Does it depend on temperature?6QM Please answer question throughly and detailed.
- One-dimensional harmonic oscillators in equilibrium with a heat bath (a) Calculate the specific heat of the one-dimensional harmonic oscillator as a function of temperature. (b) Plot the T -dependence of the mean energy per particle E/N and the specific heat c. Show that E/N → kT at high temperatures for which kT > hw. This result corresponds to the classical limit and is shown to be an example of the equipartition theorem. In this limit the energy kT is large in comparison to ħw, the separation between energy levels. Hint: expand the exponential function 1 ē = ħw + eBhw (c) Show that at low temperatures for which ħw> kT , E/N = hw(+e-Bhw) What is the value of the heat capacity? Why is the latter so much smaller than it is in the high temperature limit? Why is this behavior different from that of a two-state system? (d) Verify that S →0 as T> O in agreement with the third law of thermodynamics, and that at high T,S> kN In(kT / hw).An electron with an initial kinetic energy of 1.542 eV (in a region with 1.095 eV potential energy) is incident on a potential step (extending from x=0 to ∞) to V=2.381 eV. What is the transmission probability (in %)? FYI: If we had a travelling wave arriving at a similar potential DROP, then k1 (for x<0) would be real and the symmetry of R=(k1-k2)2/(k1+k2)2 implies reflection/transmission are the same as a potential RISE with the same energies but k1 and k2 swapped.Consider a beam of particles which is incident on a potential barrier, and assume that the potential-energy function is constant in the domains of the incident and transmitted beams. The wavefunctions representing incident, reflected and transmitted particles are given by V:(1, t) = Ae (k12–wit), V.(x,t) = Be¬i(k12+wit) and V.(x,t) = Ce(ka=–-wat), respectively. a) The current density is defined by J = 2mi Calculate the incident, reflected and transmitted current densities (J;, J,, and Jt, respectively). b) Express the transmission coefficient T = |Jt/J;| and the reflection coefficient R = |Jr/J;| in terms of A, B, C, ki and k2.
- A particle is initially prepared in the state of = [1 = 2, m = −1 >|, a) What's the expectation values if we measured (each on the initial state), ,, and Ĺ_ > b) What's the expectation values of ,, if the state was Î_ instead?Consider a weakly anharmonic a 1D oscillator with the poten- tial energy m U(x) = w?a² + Ba* 2 Calculate the energy levels in the first order in the small anharmonicity parameter 3 using TIPT and the ladder operators.The wave function of a particle at time t= 0 is given by w(0) = (4,) +|u2})), where |u,) and u,) the normalized eigenstates with eigenvalues E and E, are respectively, (E, > E, ). The shortest time after which y(t) will become orthogonal to |w(0)) is - ħn (а) 2(E, – E,) (b) E, - E, (c) E, - E, (d) E, - E,