A quantum simple harmonic oscillator can be treated as a particle of mass m attached to a spring with an effective spring constant K, oscillating between -co and +o, as we discussed in class. What is the energy of the first excited state in terms of ħ, m, and K? (b) How much energy does the particle have to absorb in order to jump from the ground state to the second excited state in terms of ħ, m, and K?
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- Consider a particle in one-dimension. In quantum mechanics, we require µ(x,t)l´ dx to be finite. Why? If this is true, we call the wavefunction admissible.An observable, q, is represented by an operator, . Assuming a system is in a state Ψ(x,t). a) Write an expression for the expectation value of the observable,<q>. b) Write an expression for the uncertainty in measuring the observable, ∆q. c) Write expressions for the operators relating to position and momentum ( and ) and for the squares of these operators. d) If Ψ(x,t) is an eigenstate of , write an equation relating the eigenvalue and itsoperator.-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=
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- a) Show that if the total energy ε of a single particle state can be written as the sum of independent energies EiA, εiB, εic... then its partition function will factorise into a product of partition functions ZAZBZC. b) Given the factorisation, show how the free energy F and quantities such as S and Cy can be expressed as a sum of terms dependent on the sources A, B, C.A qubit is in state |) = o|0) +₁|1) at time t = 0. It then evolves according to the Schrödinger equation with the Hamiltonian Ĥ defined by its action on the basis vectors: Ĥ0) = 0|0) and Ĥ|1) = E|1), where E is a constant with units of energy. a) Solve for the state of the qubit at time t. b) Find the probability to observe the qubit in state 0 at time t. Explain the result by referring to the way that the time-evolution transforms the Bloch sphere.The wavefunction for a quantum particle tunnelling through a potential barrier of thickness L has the form ψ(x) = Ae−Cx in the classically forbidden region where A is a constant and C is given by C^2 = 2m(U − E) /h_bar^2 . (a) Show that this wavefunction is a solution to Schrodinger’s Equation. (b) Why is the probability of tunneling through the barrier proportional to e ^−2CL?