Starting from the definition of the partition function, Z = Ei e-Bei, prove the following: a) (E): = b) (E2): = din Z dß 1 d² Z zdB2
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- In the canonical ensemble, we control the variables T, p, and N, and the fundamental function is the Gibbs free energy (G). But if we control T, p, and μ, then we will have a different fundamental function, Z (This is the case for cells that often regulate their temperature, pressure, and chemical potentials to maintain equilibrium). Which of the below options should the Z function equal? H - TS - μN H + TS + μN H + TS - μN G + μN F - pV - μN -H + TS + μN(c) Consider a system of two qubits with canonical basis states {|0) , |1)}. Write down an example for a two- qubit density matrix corresponding to a separable pure state and an example for a two-qubit density matrix corresponding to an entangled pure state.Example of numerical instability: Take y′ = −5y, y(0) = 1. We know that the solution should decay to zero as x grows. Using Euler’s method, start with h= 1 and compute y1, y2, y3, y4 to try to approximate y(4). What happened? Now halve the interval. Keep halving the interval and approximating y(4) until the numbers you are getting start to stabilize (that is, until they start going towards zero).
- How to evaluate the 2 partial derivatives from the expression for Z?For T = 0.3 sec, write the element of the discrete sequence corresponding to the function f(t) = 5t with k = 18. Provide your answer in seconds to the nearest first decimal place. Pay attention to the numbers givenIn the numerical example in the text, I calculated only the ratio of the probabilities of a hydrogen atom being in two different states. At such a low temperature the absolute probability of being in a first excited state is essentially the same as the relative probability compared to the ground state. Proving this rigorously, however, is a bit problematic, because a hydrogen atom has infinitely many states. Estimate the partition function for a hydrogen atom at 5800 K, by adding the Boltzmann factors for all the states shown explicitly . (For simplicity you may wish to take the ground state energy to be zero, and shift the other energies accordingly.)