Question

Consider a particle in a 3D box in which the longest wavelength transition for this system occurs at 200 nm. What is the wavelength of the transition if (a) the particle’s mass is doubled? (b) the particle’s charge is doubled? (c) the size of the box is halved?

Answer 1

for a 3D box

energy of an energy level is

E = (nx2+ny2+nz2) *h2/8 *m * a2​​​​​​

for ground state (1,1,1)

E(1,1,1) = (1+1+1)*h2/8*m*a2​​​​​​ = 3*h2/8*m*a2

for first excited state (1,1,2)or (1,2,1) or (2,1,1)

E(1,1,2) = (1+1+22)*h2/8*m*a2 = 6 *h2/8*m*a2

for highest wavelength means shortest energy, thus shortest energy transition is

changes in E = E(1,1,2)-E(1,1,1) = 3*h2/8*m*a2

in terms of wavelength,

change in E =hc/wavelength = 3*h2/8*m*a2

wavelength = 8*m*a2*c/3*h =200nm. ....... (1)(given)

(a) if particles mass is doubled, say 2m, then

then wavelength = doubled =2*200 =400 nm.

(b) if particles charge is doubled, then no effect on wavelength as in (1),no term of charge.

(c) is box is halved, say a=a/2

then in expression(1), wavength becomes 1/4th

thus wavelength becomes = 200/4 = 50nm