3. Consider the superposition state constructed of two of the above wavefunctions for the particle in a 1-D box Y(x) = b2¢2 + b3ø3 with b2 = and b3 (a) Show that the superposition state is normalized (b) Calculate the expectation value of the energy (c) List possible values of the energy that may be observed during a single measurement

3. Consider the superposition state constructed of two of the above wavefunctions for the particle in a 1-D box Y(x) = b2¢2 + b3ø3 with b2 = and b3 (a) Show that the superposition state is normalized (b) Calculate the expectation value of the energy (c) List possible values of the energy that may be observed during a single measurement

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4. Find the points of maximum and minimum probability density for the nth state of a particle in a one- dimensional box. Check your result for the n=2 state.
3. Particle in a 2D Box. A quantum mechanical particle is confined in side a square 2D box, with side length L. Inside the box V=0 and outside the box V=infinity. Let the wave function to be (x,y). (a) write down the Schrodinger equation of (x,y). (b) Use the separation of variable method solve (x,y) (let the quantum numbers to be nx and ny.) (c) What is the energy for the state (nx, ny)? (d) What is the probability density p(x,y) for the state nx=3 and ny=3? Sketch this p(x,y) in a square.
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.