Quantum Mechanics of a Harmonic Oscillator: Eigenstates and Solutions
Explore the eigenstates and solutions of the harmonic oscillator in quantum mechanics, covering topics such as solving differential equations, the operator formalism, Hermite polynomial solutions, and relationships within the system. Understand the representation of position and momentum operators in terms of energy eigenstates.
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PHY 712 Quantum Mechanics 12-12:50 PM MWF Olin 103 Plan for Lecture 8: Start reading Chapter #7 in Shankar; Eigenstates of the harmonic oscillator 1. Solution of the differential equation 2. Operator formalism 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 1
9/13/2017 PHY 741 Fall 2017 -- Lecture 8 2
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9/13/2017 PHY 741 Fall 2017 -- Lecture 8 4
Quantum mechanics of a harmonic oscillator https://en.wikipedia.org/wiki/Simple_harmonic_motion from m k x Classical trajectory: k m ( ) = = ( ) cos where x t X t 0 Oscillator potential: 1 ( ) 2 1 2 = 2 2 2 V x kx m x 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 5
V(x)=1/2 k x2 x Comes from analysis of systems near equilibrium: 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 6
Eigenstates of the the Schrdinger equation: Solutions to the differential equation -- let 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 7
In these units, the equation becomes: Further transformation: Equation for u(y): Hermite polynomial solutions for u(y)=Hn(y) 2 2 y dy dy d H d H + = 2 0 n n nH n 2 1 2 ( ) 1 2 = = + 2 ( ) y 0 n H n n n 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 8
Complete solution including normalization ( ) nx 1 2 = + = 0,1,2,..... E n n n 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 9
Some useful relationships of Hermite polynomials: Useful matrix elements: 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 10
Representation of the position and momentum operators in terms of the energy eigenstates of the harmonic oscillator: n= 0 1 2 3 .. Note that: 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 11
Average values and uncertainties ( ) = + 2 0 = 2 1 n X n n X n n 2 m m ( ) = + 2 0 = 2 1 n P n n P n n 2 ( ) X n n p n + = 2 1 n n 2 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 12
Analysis of the Harmonic Oscillator Hamiltonian in terms of raising and lowering operators Define: It follows that: 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 13
Note that: Unitless operators: 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 14
Eigenstates of Note that: 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 15
= 1 a C = + 1 a D Argue that the eigenvalues have a sequence and can be labeled with energy n=0,1,2,3 0 a = 0 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 16
This implies 1 2 = + Where n H n n n n Determining coefficients: a = = + 1 C 1 a D 1 2 = = H n a a n n n 2 = = = 1 1 n a a n an an C n n n n n n = = + similarly, 1 C n D n n n 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 17
Summary of results 1 2 1 2 = + = + H n a a n n n = 1 a n n n = + + 1 1 a n n n 9/13/2017 PHY 741 Fall 2017 -- Lecture 8 18
