Other examples of one-dimensional motion PHY 741 Quantum Mechanics 12-12:50 PM MWF Olin 103 Plan for Lecture 9: Continue reading Chapter #7 in Shankar; Eigenstates of the one-dimensional Schrödinger equation Harmonic oscillator Comment on HW #5 Other examples of one-dimensional motion 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

9/15/2017 PHY 741 Fall 2017 -- Lecture 9

9/15/2017 PHY 741 Fall 2017 -- Lecture 9

One-dimensional harmonic oscillator Define: Note that: 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

Representation of the position and momentum operators in terms of the energy eigenstates of the harmonic oscillator: n= 0 1 2 3 ….. 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

Representation of the raising and lowering operators in terms of the energy eigenstates of the harmonic oscillator: 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

9/15/2017 PHY 741 Fall 2017 -- Lecture 9

Example from statistical mechanics: 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

Example: Particle of mass m confined within an finite square well: Comment on HW #5 Example: Particle of mass m confined within an finite square well: V(x) V0 V0 x a -a 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

After some algebra, we found that the energy eigenvalues E were solutions of transcendental equations: For even parity For odd parity 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

Solution using maple: 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

Another example of a one-dimensional system Consider an electron moving in a one-dimensional model potential (Kronig and Penney, Proc. Roy. Soc. (London) 130, 499 (1931) V0 a 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

V0 a 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

V0 a 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

V0 a Band gap E(k) 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

Consider a single potential well Spectrum 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

Consider a periodic potential well system Spectrum Consider a periodic potential well system E V0 a 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

V0 a b 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

V0 a b 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

V0 a b 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

Forbidden states f Forbidden states v 9/15/2017 PHY 741 Fall 2017 -- Lecture 9

f Band gap Band gap ka/p 9/15/2017 PHY 741 Fall 2017 -- Lecture 9