Given this quantum state: | ¥(r,0,0) = R(r) (+Y+/Y ¹² — — Y₂²), a) measured: |Z|² find possible outcomes, corresponding probabilities and
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- Consider the electron-hole overlap integral Mnn for a quantum well given by: Mn Pen (2) Pnn (z) dz. %3D - 00 n' and zerd (i) Show that Mon is unity if n otherwise in a quantum well with infinite barriers. (ii) Show that Mon is zero if (n-n') is an odd number in a quantum well with finite barriers.(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?For a quantum particle in a scattering state as it interacts a certain potential, the general expressions for the transmission and reflection coefficients are given by T = Jtrans Jinc R = | Jref Jinc (1) where Jinc, Jref, Jtrans are probability currents corresponding to the incident, reflected, and transmitted plane waves, respectively. (a). potential For the particle incident from the left to the symmetric finite square well -Vo; a < x < a, V(x) = 0 ; elsewhere, show that B Ꭲ ; R = A A
- A deuterium molecule (D2₂) at 30°K is known to be in the state, 1 /26 12/₂) = = |3|1, 1) + 4 |7, 3) + |7, 1) where , m) are eigenstates of the angular momentum operator. (a) If one were to measure L₂, what posible values one would get and what would be their associated probabilities? (b) Repeat (a) but for L². (c) What is the expectation value of the energy (E) of the molecule in this state, assuming purely rotational states. Take c= 30.4 cm-¹, where I=moment of inertia of D₂ and c=speed of light. Express your answer in eV. -Consider a state of a 2-electron diatomic molecule AB described by the electronic normalized wave function Þ(1, 2) = y(1,2) [a(1) B(2) – a(2) B(1)] where p(1, 2) is the spatial part of the electronic wave function. (a) What must be the value of the integral (p(1, 2)|4(1,2)) so that the complete (spatial and spin) function (1, 2) is normalized?. (b) Is the spatial function p(1, 2) symmetric or antisymmetric with respect to the exchange of the space coordinates of electron 1 and 2?