An electron is confined in one dimensional atomic box of length, L = 5 * 10^- m.

Question

An electron is confined in one dimensional atomic box of length, L = 5 * 10^- m. The energy levels are set as energies of particle in the box. (m_c = 9.1 times 10^31 kg = 6.626 times 10^-34 J middot s) a) What is the ground state (n=1, the first energy state) of this electron? b) If the electron can be excited to n=2 and n=3, states. How much energy has to be absorbed for electron to climb from n=2 to n=3 state? c) If photon is emitted when the excited electron return to lower state. What is the frequency and the wave length of the photon, when electron returns from n=2 to n=1 state?

Explanation / Answer

given, length of the box, L = 5*10^-10 m

me = 9.1*10^-31 kg

h = 6.62*10^-34 Js

a) wavelength of the electron in the gound state = 2L

form de broglie's wavelength formula, assuming velocity of electron in the vwell to be v

2L = h/mv

v = h/2mL = 6.62*10^-34/2*9.1*10^-31*5*10^-10 = 7.27*10^5 m/s

KE = 0.5mv^2 = 0.5*9.1*10^(-31)*v^2 = 2.4079*10^-19 J

b) so wavelength at n = 1 is 2L

at n = 2 is L

at n = n is 2L/n

2L/n = h/mv

v = nh/2Lm

E = KE = 0.5mv^2 = 0.5*m*n^2h^2/4L^2m^2 = n^2*h^2/8mL^2

so E1 = 2.4079*10^-19 J

En = n^2E1

so E2 = 9.6317*10^-19 J

E3 = 2.167*10^-18 J

energy required to go from n = 2 to n = 3 = E3 - E2

E(3-2) = 12.0383*10^-19 J

c) energy differnec ein n = 2 and n = 1 states = E2 - E1 = 7.2238*10^-19 J

let the photon frequency be mu

then h*mu = 7.2238*10^-19 = 6.62*10^-34*mu

mu = 1.091*10^15 Hz

wqavewlength , lambda = c/mu = 2.749*10^-7 m [ here c is speed of light]

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