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- -h? d? 3. Find average value of kinetic energy, for ground state of the harmonic 2µ dx? oscillator. The unnormalized wavefunction of oscillator is given by: y,(x)= e k Express your answer in terms of the oscillation frequency, @= where a=2) Consider a particle in a three-dimensional harmonic oscillator potential V (r, y, z) = 5mw²(r² + y² + z®). The stationary states of such a system are given by ntm(r, y, z) = vn(x)¢r(y)v'm(2) (where the functions on the right are the single-particle harmonic oscillator stationary states) with energies Entm = hw(n +l+m+ ). Calculate the lifetime of the state 201.Problem : Given the transfer function of a linear lung mechanics model S G(s) s3 + 3s2 + 2s a. Find the full-dimensional (3-D) controllable realization. b. Find a minimal realization. c. Verify the minimal realization is both controllable and observable.
- Consider the function v(1,2) =( [1s(1) 3s(2) + 3s(1) 1s(2)] [x(1) B(2) + B(1) a(2)] Which of the following statements is incorrect concerning p(1,2) ? a. W(1,2) is normalized. Ob. The function W(1,2) is symmetric with respect to the exchange of the space and the spin coordinates of the two electrons. OC. y(1,2) is an eigenfunction of the reference (or zero-order) Hamiltonian (in which the electron-electron repulsion term is ignored) of Li with eigenvalue = -5 hartree. d. The function y(1,2) is an acceptable wave function to describe the properties of one of the excited states of Lit. Oe. The function 4(1,2) is an eigenfunction of the operator S,(1,2) = S;(1) + S,(2) with eigenvalue zero.Q3/1 Find the difference State in Second and first excited energies (AE) of Particle 1-D with length (L).Subject: Mathematical Physics Topic: Functions of a complex variable.
- 2. Consider a system with Neumann boundary conditions. Show that the Neumann Green's function is symmetric under exchange of its two position variables i.e. Gr(7,7") = Gy(", 5). GN(F,")A particle is described by the wavefunction Ψ(t, x), and the momentum operator is denoted by pˆ. a) Write down an expression for the differential operator pˆ. b) Write down an expression for the expectation value of the momentum, ⟨p⟩. c) Write down an expression for the probability density, ρ. d) Write down an expression for the probability of finding the particle between x = a and x = b.6. (a) Show how the one-dimensional time-dependent Schroedinger equation d?y(xt)/dx? = -Bd²w(xt)/dt? can be reduced to a one-dimensional time-independent Schroedinger equation d? y(x)/dx? = -B y(x) using for the time-dependent wavefunction y(x,t), the following expression: y(&t) = ylx)cos(@ot) %3D Hint: equate the 2nd derivative with respect to x to that with respect to t, then simplify. (b) also, determine the expression for ß from the resulting expression in (a).
- 4. For the potential if 0 a. calculate the product of the uncertainties O„0, for the second excited state V3(x) : sin a aConsider a potential barrier represented as follows: U(x) = 0 if x < 0; εx if 0 < x < a; 0 if x > a Determine the transmission coefficient as a function of particle energy.Determine the expectation values of the position (x) (p) and the momentum 4 ħ (x)= cos cot,(p): 5V2mw 4 mah 5V 2 sin cot 2 ħ moon (x)= sin cot, (p)= COS at 52mo 2 4 h 4 moh (x)= 52mo sin cot.(p) COS 2 h s cot, (p) 5V2mco 2 moh 5V 2 sin of as a function of time for a harmonic oscillator with its initial state ())))