2-1 Dr. Vasileios Lempesis PHYS 454 HANDOUT 2-Schroedinger equation 1. Show that the average value of the momentum p is zero in the () following two cases: a) when the wavefunction ! x is real and b) () when ! x is even or odd. "# x 2. The wavefunction of a particle has the form ! (x) = Ne . a) Find N. b) What is the probability of finding the particle in the region !1 " x " 1 if we consider ! = 1 ? 3. Show that the average value of the square of the momentum in an one dimensional problem could always be written in the form +# p 2 =h 2 $ ! (x) ' 2 dx . "# ( )( ) 4. Calculate the product of uncertainties !x !p for the Gaussian function () ! x = 4 " $ " x2 / 2 e . # 5. Calculate the uncertainty in momentum and position for a particle with a wavefunction equal to () ! x = Nxe" # x 2 /2 . 6. Find the average value of the momentum for any wavefunction of the form () () ! x = " x eikx () where ! x is a real and square integrable wavefunction. 7. Let Pab (t) be the probability of finding the particle in the range ( a < x < b) , at time t. Show that dPab = J a,t ! J (b,t) dt ( ) where 2-2 Dr. Vasileios Lempesis J (x,t) ! ih % #" * #" ( . " $ "* ' 2m & #x #x *) What are the units of J? [ J (x,t) is called the probability current, because it tells you the rate at which probability is “flowing” past the point x. If Pab (t) is increasing, then more probability is flowing into the region at one end than flows at the other.] 8. A particle is represented (at time t=0) by the wave function ( ) $& A a 2 " x 2 , ! x,0 = % &' 0 ( ) "a#x#a otherwise. (a) Determine the normalization constant A. (b) What is the expectation value of x (at time t=0)? (c) What is the expectation value of p (at time t=0)? (d) Find the expectation value of p 2 . (e) Find the uncertainty in x. (f) Find the uncertainty in p. (g) Check that your results are consistent with uncertainty principle. 9. A particle of mass m is in the state () ! x = Ae ( ) " a #% mx 2 / h + it &( $ ' (a) Find A. (b) For what potential energy function V (x) does ! satisfy the Schroedinger equation? (c) Calculate the expectation values for x, p, p 2 and x 2 . (d) Find the uncertainties in x and p. Are they consistent with the uncertainty principle?