Question

1) (35 points) The wave function for a particle moving along x axis between the limits 0 and L is: (x)-C sin (nx xL) where n are 1, 2, 3, A) Determine the normalization constant C B) Why can't n take the value of 0, briefly explain C) For n-3 determine the values of x (in terms of L) that correspond to a maximum or a minimum in the wave function D) For n-3 determine the values of x (in terms of L) that correspond to a node. Use what you need of the following: d/dx (sin ax) - a cos ax sin(ax) so, then axen'π where n'= 0, 1, 2, 3, . cos(ax) :0, then ax,t/2 + n,π where n's 0, 1, 2, 3, d/dx (cos ax)- -a sin ax sin2 (ax) dx = x/ 2-14a (sin 2ax) 2) (30 points) The energy of quantum states in the particle in a box of length Lis: 8mL Where n 1, 2, 3,... m is the mass of the particle A) Derive the equation for the energy difference between successive states (n) and (n+1) B) Estimate the energy difference between the two energy states 2 and 3 for an electron confined in a box with the dimension L of 15 nm (nanometer 1X10*m); Mass of the electron 9.109 x 102 g Determine the wavelength of the light required to achieve the excitation of the electron C) from state 2 to 3. (Recall E-h C/A) where C is speed of light (3X10* m.sec (10 points) The operator P, that is used to determine the momentum of a particle in a box is 3) h/i (d/dx). The wave function for a particle moving along x axis between the limits 0 and Lis (x)-Csin (nn x/L) Show whether Wn (x) is an eigenfunction of P Given: d/dx (sin ax)- a cos ax

Answer 1

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