Chapter 4 Free and Confined Electrons Lecture given by Qiliang Li Dept. of Electrical and Computer Engineering George Mason University ECE 685 Nanoelectronics

§ Parabolic Well – Harmonic Oscillator

The Schrodinger’s equation is:

§ Parabolic Well – Harmonic Oscillator Let divergent

§ Parabolic Well – Harmonic Oscillator Let

§ Parabolic Well – Harmonic Oscillator Therefore, H(x) grows like exp(x^2), producing unphysical diverging solution. So the coefficients beyond a given n should vanish, the infinite series becomes a finite polynomial. So we should have:

§ Parabolic Well – Harmonic Oscillator n is a non-negative integer: 0, 1, 2, …

§ Parabolic Well – Harmonic Oscillator ladder operator

§ Parabolic Well – Harmonic Oscillator Use ladder operator to find the wave function:

§ Parabolic Well – Harmonic Oscillator Let: Only a 1 is not 0: Similarly, we can find more wavefunction…

§ Parabolic Well – Harmonic Oscillator In (A-B), the particle (represented as a ball attached to a spring) oscillates back and forth. In (C-H), some solutions to the Schrödinger Equation are shown, where the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. (C,D,E,F), but not (G,H), are energy eigenstates. (H) is a coherent state, a quantum state which approximates the classical trajectory.

§ Triangular Well

Example:

§ 4.6 Electron confined to atom See lecture note