Ground state in an infinite well - Example

1. An electron is confined to a 1 micron sized piece of silicon. Assuming that the semiconductor can be adequately described by a one-dimensional quantum well with infinite walls, calculate the lowest possible energy within the material in units of electron volt. If the energy is interpreted as the kinetic energy of the electron, what is the electron velocity? (The effective mass of electrons in silicon is 0.26 m₀, where m₀ = 9.11 x 10^-31 kg is the free electron rest mass).
   Answer: Starting from the expression for the energy levels in an infinite quantum well:
   
   with n = 1 to find the lowest bound state energy one finds:
   E (eV)= 1/q h²/ (8m_e^* L²)
   = 1/(1.6 x 10^-19) x (6.625 x 10^-34 / 10^-6)² / (8 x 0.26 x 9.11 x 10^-31) = 1.44 meV
   As this energy corresponds to k = p/L and p = m_e^* v = (h/2p) k the velocity is obtained from:
   v = p/m_e^* = h/(2L m_e^* ) = 1400 m/s = 1.4 x 10⁵ cm/s.