Show that the dispersion relation for the lattice vibrations of a chain of identical masses M, in which each is connected to its first and second nearest neighbours by springs of spring constants K and K, respectively, is Mo = 2K[1- cos(ka)]+2K,[1-cos(2ka)] where a is the equilibrium spacing. Show that: this dispersion relation reduces to that for sound waves in the long-wavelength limit (ensure that the velocity corresponds to that predicted by the elastic modulus of the crystal); the group velocity vanishes at k = t7/a; and o is periodic in k with period 2/a.
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- NoneLike a harmonic oscillator with a force constant of 1550 N/m of the nitrogen oxide molecule suppose you behave. The energy of the second excited vibrating state (in eV) Find.A particle on a ring has the following wave function: Y=1/(2n) e-7iø What is the smallest value of where the first node appear in the imaginary part of the wave function (excluding -0). Give your answer as an angle in units of radians (remember that 1 rad = 57.3 degrees, and 2π rad = 360 degrees).
- For quantum harmonic insulators Using A|0) = 0, where A is the operator of the descending ladder, look for 1. Wave function in domain x: V(x) = (x|0) 2. Wave function in the momentum domain: $(p) = (p|0)Calculate the phase shift 8 (K) for the S-wave scattering (1-0). Assume that the potential is given as the delta - Shell 2 U (0)_ _h² 2m S(x-a)6QM Please answer question throughly and detailed.
- The dispersion relation for one dimensional lattice vibrations of chain of identical mass m, in whach mass is connected to first and second nearest neighbors by coupling constants C: and Cis me =C,(a-cos(nka). For ka c«1, the speed of sound in this limit has fom 4C,+C ka C+2C, ka m 2C, +C, ka C+4C, ka mConsider a particle moving in a one-dimensional box with walls at x = -L/2 and L/2. (a) Write the wavefunction and probability density for the state n=1. (b) If the particle has a potential barrier at x =0 to x = L/4 (where L = 10 angstroms) with a height of 10.0 eV, what would be the transmission probability of the electrons at the n = 1 state? (c) Compare the energy of the particle at the n= 1 state to the energy of the oscillator at its first excited state.Suppose that the ground vibrational state of a molecule is modelled by using the particle-in-a-box wavefunction ψ0 = (2/L)1/2 sin(πx/L) for 0 ≤ x ≤ L and 0 elsewhere. Calculate the Franck–Condon factor for a transition to a vibrational state described by the wavefunction ψ′= (2/L)1/2sin{π(x −L/4)/L} for L/4 ≤ x ≤ 5L/4 and 0 elsewhere.
- An arbitrary quantm mechanical system is initially in the ground state |0). At t = 0, a perturbation of the form H' (t) = Hoc:/T is applied. Show that at large times the probability that. tlhe system is in state |1) is given by |(0}Ho|1}|2 (A) A + (Ac)? where As is the difference in cnergy of states |0) and |1). Be specific about what assumption, if any, were inade arriving at your conclusion.15 6x4 – 12Lx³ + 15 L²x? - 3,3 (- L²r? L3x 6r4 - N2 2me 12La3 dx = dx 2 2 2 Evaluate the integral in the numerator. p*H@ dx = N² 40m h² L³ e 40me The denominator of the E expression from Step 1 is 6 _ 3Lx + 13 ,2 4 -L²x 4 Lx³ + F8-3Lz' + 12rt - (%) Lz3 + +L교2 p dx = N² dx Evaluate the integral in the denominator. 1 1 φ φ αχx = N2 840 840 Step 3 of 6 Divide the numerator by the denominator (both from Step 2) and simplify. (Use the following as necessary: ħ, L, me, P, T, and x.) 21h? E L²m 21h? L²me e Step 4 of 6 Calculate the energy for an electron in a 0.43-nm box using the formula from Step 3. = 4.0 0.26 X 840 kJ•mol-1 φ Step 5 of 6 Calculate the exact energy for an electron in the first excited state in a 0.43-nm box. n2h2 Recall that for a particle in a one-dimensional box we can write En we can therefore calculate an exact solution. 8mL2' Eexact = 4.0 197 X kJ-mol-1 Submitpls answer d and e