Examples and Exercises

Normalizing a Wavefunction Find a normalizing factor of the hydrogen’s electron wavefunction Function of r, using spherical coordinate

Eigen eqaution Show if e ax is an eigenfunction of the operator d/dx Show if e ax 2 is an eigenfunction of the operator d/dx Operator Constant Function

Orthogonal wavefunction Both sinx and sin2x are eigenfunction of d/dx, show if sinx and sin2x are orthogonal.

Expectation Value Calculate the average value of the distance of an e - from the n of H-atom

Uncertainty Principle Calculate the minimum uncertainty in the position of mass 1.0 g and the speed is known within 1  m s -1. Uncertainty in position along an axis Uncenrtianty in linear momentum

Probability (Particle in a box) Wave function of conjugated electron of polyene can be approximated by PAB. Find the probability of locating electron between x=0 and x=0.2 nm in the lowest state in conjugated molecule of length 1.0 nm when n=1 L=1.0 nm and l = 0.2 nm

Harmonic Oscillator Find the normalizing factor of Harmonic Oscillator wavefunction

Harmonic Oscillator The bending motion of CO 2 molecule can be considered as a harmonic oscillator, find the mean displacement of the oscillator

Exercises Calculate the speed of an electron of wavelength 3.0 cm Calculate the Brogile wavelength of a mass of 1.0 g travelling at 1.0 cm s-1 Calculate the probability of a particle in ground state between x=4.0 and 5.0 cm in a box of 10.0 cm length Calculate the probability of a hydrogen’s electron in ground state to be found within radius a 0 /2 from the nucleus

Exercises Identifiy which functions are eigenfunctions of the operator d/dx e ikx coskx e -ax 3 Calculate the energy separation between the levels n=2 and n=6 of an electron in a box of length 1.0 nm What are the most likely locations of a particle in a box of length L in the state n=3?

Exercises What are the most likely locations of a particle in a harmonic oscillator well of lin the state =5 ? Confirm that the wavefunction for the ground state of a one- dimention linear harmonic oscillator is a solution of the Schrödinger eqaution Write down the Harmonic Oscillator wavefunction in the state =0 and 4 Write down the Rigid Roter wavefunction Y 0,0, Y 1,2, Y 2,1 and Y 2,-2 and calculate their energies