1. A quantum system goes into a time-dependent superposition of three real eigen- functions (ener...
1. A quantum system goes into a time-dependent superposition of three real eigen- functions (energies E, E 2, E, all equally likely). Write down the total wave- function, and calculate the probability density. Express your answer in terms of the (co)sinusoidal interference terms. 2. Write down the time-dependent wavefunction for the particle a box that is in a superposition of the n = 2 and n = 4 states. Assume there is a 30% chance of measuring state n 2 3. Calculate the probability density for this wavefunction, and determine the os- cillation frequency between states (recall that w = 4. Suppose an electron-positron pair is produced from the vacuum. Recalling that E = mc2, determine the extent of time they can exist before annihilating if (a) They are at rest (b) They are travelling at -0.5c. 5. Using Maple (or Mathematica...), show that the quantum harmonic oscillator wavefunctions 2hH.,(x) are indeed orthogonal, i.e. only when m = n. Type with (orthopoly) in Maple to load the functions from the lbrary. Call them in the code as H(n, x). Remember to set m,w, and to 1 before attempting numerical integration! (b) Show that the orthogonality condition is true regardless of the bounds of integration, so long as the integral is evaluated symmetrically about the origin i.e. from-a to a for arbitrary a)
1. A quantum system goes into a time-dependent superposition of three real eigen- functions (energies E, E 2, E, all equally likely). Write down the total wave- function, and calculate the probability density. Express your answer in terms of the (co)sinusoidal interference terms. 2. Write down the time-dependent wavefunction for the particle a box that is in a superposition of the n = 2 and n = 4 states. Assume there is a 30% chance of measuring state n 2 3. Calculate the probability density for this wavefunction, and determine the os- cillation frequency between states (recall that w = 4. Suppose an electron-positron pair is produced from the vacuum. Recalling that E = mc2, determine the extent of time they can exist before annihilating if (a) They are at rest (b) They are travelling at -0.5c. 5. Using Maple (or Mathematica...), show that the quantum harmonic oscillator wavefunctions 2hH.,(x) are indeed orthogonal, i.e. only when m = n. Type with (orthopoly) in Maple to load the functions from the lbrary. Call them in the code as H(n, x). Remember to set m,w, and to 1 before attempting numerical integration! (b) Show that the orthogonality condition is true regardless of the bounds of integration, so long as the integral is evaluated symmetrically about the origin i.e. from-a to a for arbitrary a)