According to the equipartition theorem of classical thermodynamics, all ocillators in the cavity have the same mean energy, irrespective of their frequencies. With this in view, prove the following relation: 00 EE¬E/kT dE < E >= = kT . S" e-E/KT dE
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- Please don´t answer by partition equation!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.Show that the Maxwell speed distribution function F(v) approaches zero by taking the limit as v → 0 and as v → ∞
- Derive the following general equation,which is similar to the Gibbs-Helmholtz equation:(∂/∂T)(F/T)=−U/T^2Given the internal energy U and entropy S of N weakly interacting particles in a closed system with fixed volume V. U = NkgT² (27In 2) U S = Nkg lnz + T (a) Prove the Helmholtz free energy (b) Prove the Pressure of the system is F = -NkBT ln z P = Nk Tln z) ( TWhat does population vector, Π=(P1,P2,P3r,P3w)T mean ? How do this formula describe the overall state probability? (there are state 1, state 2 and state 3w, 3r)
- 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 + μNShow 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.How to solve this question
- The Hamiltonian for classical ideal gas for N atoms is H = P? (i) Calculate E(E) and obtain the following entropy relation (4rm E 3 S(E,V) = Nklog +-Nk 3h? N (ii) Calculate internal energy U(S,V). (ii) Calculate specific heat capacity.how do I calculate an average energy when there are three psi unknowns? PSI= c1*psi1(x) + c2*psi2(x) + c3*psi3(x) also, how do I calculate probability when there is an energy given?