Quantum Mechanics

From Principles of Electronic Materials and Devices, Third Edition, S From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

MOTION OF AN ELECTRON IN ONE DIMENSIONAL POTENTIAL WELL (PARTICLE IN A BOX) Consider an electron of mass m, moving along positive x-axis between two walls of infinite height, one located at x=0 and another at x=a. Let potential energy of the electron is assumed to be zero in the region in-between the two walls and infinity in the region beyond the walls. X-axis X=0 X=a V= V=0

Region beyond the walls: The Schrodinger’s wave equation representing the motion of the particle in the region beyond the two walls is given by The only possible solution for the above equation is ψ=0. Since ψ=0 , the probability of finding the particle in the region x<0 and x>a is Zero . i.e., particle cannot be found in region beyond the walls. Region between the two walls: The Schrodinger’s wave equation representing the motion of the particle in the region between the two walls is given by

Solution of the equation 1 is of the form Where A and B are unknown constants to be determined. Since particle cannot be found inside the walls

The equations are called boundary conditions The equations are called boundary conditions. Using the I boundary condition in equation 2, we get Therefore equation 2 becomes Using condition II in equation 3 we get

Therefore correct solution of the equation 1 can be written as The above equation represents Eigenfunctions. Where n=1,2,3,.. (n=0 is not acceptable because, for n=o the wavefunction ψ becomes zero for all values of x. Then particle cannot be found anywhere) Substituting for  in equation 1a we get Therefore energy Eigenvalues are represented by the equation

Particle in the Infinite Potential Well

Normalization of wave function: Where n=1,2,3,.. It is clear from the above equation that particle can have only desecrated values of energies. The lowest energy that particle can have corresponds to n=o , and is called zero-point energy. It is given by Normalization of wave function: We know that particle is definitely found somewhere in space Therefore Normalised wavefunction is given by

PHYSICAL INTERPRETATION OF WAVE FUNCTION The state of a quantum mechanical system can be completely understood with the help of the wave function ψ. But wave function ψ can be real or imaginary. Therefore no meaning can be assigned to wavefunction ψ as it is. According to Max Born’s interpretation of the wavefunction, the only quantity that has some meaning is which is called as probability density.

Thus if ψ is the wavefunction of a particle within a small region of volume dv, then gives the probability of finding the particle within the region dv at the given instant of time. We know that electron is definitely found somewhere in the space. The wavefunction ψ, which satisfies the above condition, is called normalized wavefunction. dv

Similarly for n=2   x a

  Wavefunctions, probability density and energies are as shown in the figure. n=1 n=2 1 P1 2 P2 x=0 x=a/2 x=a

Free Particle   Consider a particle of mass m moving along positive x-axis. Particle is said to be free if it is not under the influence of any field or force. Therefore for a free particle potential energy can be considered to be constant or zero. The Schrodinger wave equation for a free particle is given by.

The solution of the equation 1 is of the form Where A and B are unknown constants to be determined. Since there are no boundary conditions A, B and  can have any values. Energy of the particle is given by Since there is no restriction on  there is no restriction on E. Therefore energy of the free particle is not quantised. i.e., free particle can have any value of energy.

Probability to Find particle in the Right Half of the Well

Average Momentum of Particle in a Box (Infinite Potential Well) Can be shown that Reflects the vector nature of the momentum a particle have the same probability of moving to the right or to the left