10. A particle is represented by the following wave function: (x) = 0 = C(2x/L + 1) C(-2x/L + 1) = = 0 x <-L/2 -L/2 +L/2 (a) Use the normalization condition to find C. (b) Evaluate the probability to find the particle in an interval of width 0.010L at x = L/4 (that is, between x = 0.245L and x = 0.255L. (No integral is necessary for this calculation.) (c) Evaluate the probability to find the particle between x = 0 and x = +L/4. (d) Find the average value of x and the rms value of x: rms=√(2²) av

A real wave function is defined on the half-axis: [0≤x≤00) as y(x) = A(x/xo)e-x/xo where xo is a given constant with the dimension of length. a) Plot this function in the dimensionless variables and find the constant A. b) Present the normalized wave function in the dimensional variables. Hint: introduce the dimensionless variables = x/xo and Y(5) = Y(5)/A.

10. A particle is represented (at time t = 0) by the wave function ¥(x,0) = {4(a² ¯ 0, JA(a²-x²), if- a ≤x≤+a otherwise (a) Determine the normalization constant A. (b) What is the expectation value of x (at time t = 0)? d (c) What is the expectation value of p (at time t = 0)? (Note that you cannot get it from p = m² .Why dt not?) (d) Find the expectation value of x². (e) Find the expectation value of p².

3. A system of N harmonic oscillators of frequency w are prepared in identical initial states of wavefunction Þ(x,0). It is found that the measurement of the energy of the system at t = 0 gives 0.5hw with probability 0.25, 1.5hw with probability 0.5 and 2.5hw with probability 0.25. a. Write a possible function p(x, 0). b. Write the corresponding (x, t). c. What is the expectation value of the Hamiltonian in the state p(x,t) ? d. Calculate the expectation value of position at time t