The position as a function of time x(t) of a simple harmonic oscillator is given by: x(t) = A cos(wt) where A is the amplitude and w is the angular velocity. a) What is the range of possible values of x permitted for this oscillator? b) Derive the probability density function of p(x) for this oscillator. c) Validate that p(x) is normalized.

A particle confined in a one-dimensional box of length L(<= X <= L) is in a state described by the wave function where **check attached image** where A and B are constants given by real numbers. A) Determine which relationship A and B must satisfy for the wavefunction to be normalized. B) Suppose that A = B. What is the probability of the particle being found in the interval 0 <= X <= L/2? C) What are the values ​​of A and B that minimize the probability of finding the particle in the range of positions 0 <= X <= L/2?

At t = 0, a particle is prepared to assume the state V(x,0) = ¹+2i₁(x) + cos (²) sin( ³ )&₂(x) + A¥3(x) where A is a constant and the eigenfunctions ₁(x), 2(x), and 3(x) correspond to energies E₁, E2, and E3, respectively. What is the probability that the particle has energy E3? a. 0.822 b. 0.325 c. 0.675 d. 0.570

QUESTION 1 The expectation value is the strict average of the possible values. O True False

One-dimensional harmonic oscillators in equilibrium with a heat bath (a) Calculate the specific heat of the one-dimensional harmonic oscillator as a function of temperature. (b) Plot the T -dependence of the mean energy per particle E/N and the specific heat c. Show that E/N → kT at high temperatures for which kT > hw. This result corresponds to the classical limit and is shown to be an example of the equipartition theorem. In this limit the energy kT is large in comparison to ħw, the separation between energy levels. Hint: expand the exponential function 1 ē = ħw + eBhw (c) Show that at low temperatures for which ħw> kT , E/N = hw(+e-Bhw) What is the value of the heat capacity? Why is the latter so much smaller than it is in the high temperature limit? Why is this behavior different from that of a two-state system? (d) Verify that S →0 as T> O in agreement with the third law of thermodynamics, and that at high T,S> kN In(kT / hw).

The wave function W(x,t)=Ax^4 where A is a constant. If the particle in the box W is normalized. W(x)=Ax^4  (A x squared), for 0<=x<=1, and W(x) = 0 anywhere. A is a constant. Calculate the probability of getting a particle for the range x1 = 0 to x2 = 1/3 a. 1 × 10^-5 b. 2 × 10^-5 c. 3 × 10^-5 d. 4 × 10^-5

A particular two-level quantum system has two eigenstates given by Ig) := (9) le) := (¿) where |g) is the ground state and le) is the excited state. The Hamiltonian operator is given by Îû = () O A a. Write down the density matrix and calculate for the partition function. b. Calculate for the mean energy of the system. c. Calculate for the expectation value of the operator 1 0 0-1

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.