PHYS 773: Quantum Mechanics February 6th, 2012 Algebraic Method of Solving the Linear Harmonic Oscillator Lecture by Gable Rhodes Refer to Merzbacher, p220-224 and Griffiths, p40-51 PHYS 773: Quantum Mechanics February 6th, 2012

Simple Harmonic Oscillator Many physical problems can be modeled as small oscillations around a stable equilibrium Potential is described as parabolic around the minimum energy The Hamiltonian is formulated in the usual way for canonical variables q and p. Gable Rhodes, February 6th, 2012

Raising & Lowering Operators - Defined After substituting in our potential, we can rearrange the constants slightly to give the Hamiltonian a more suggestive appearance If we “factor” the operators, we get Recognizing that the last term is the commutator of the canonical variables, we get Gable Rhodes, February 6th, 2012

Raising & Lowering Operators - Defined We can now define the operator a Upon substitution And Simplifying Gable Rhodes, February 6th, 2012

Raising & Lowering Operators - Defined If we reverse the order of the operators, a similar expression is obtained Subtracting the two forms yields the commutator Simplifying to Gable Rhodes, February 6th, 2012

Raising & Lowering Operators - Defined It is important to note that the a, a† operators are defined in such a way that as long as the state variables follow the canonical commutation relation, the a, a† commutator will be 1. And the Hamiltonian can be written as a linear function of a, a† Gable Rhodes, February 6th, 2012

Properties of a, a† If a wave function has the property that it is an eigenfunction of the Hamiltonian We can introduce the equivalent statement for the operator aa† with generic eigenvalue, λ We then apply the a† operator to a†ψ and test if the result is the same eigenvector. Use commutator Gable Rhodes, February 6th, 2012

Properties of a, a† And the equivalent method for a Using the relationship of Hamiltonian, we can then relate eigenvalues Gable Rhodes, February 6th, 2012

Raising & Lowering Operators -Properties a† is called the raising (or creation) operator. And a is the lowering (or annihilating) operator. The rungs of the ladder are all evenly separated. No degeneracy. a† a† ψ λ+2 a† ψ λ+ 1 ψ λ aψ λ- 1 aaψ λ-2 Wave vectors Eigenvalues Gable Rhodes, February 6th, 2012

What is the Significance? If we have any solution, we can find infinitely more solutions by repeated application of the raising and lowering operators But, importantly, although there are infinite solutions we know that the are are no solutions with negative energy (both kinetic and potential components for the Hamiltonian are always positive) Therefore, a ground state must exist. (this is also a property of Sturm-Liouville PDE) Applying the lowering operator to the ground state will result in a null vector (trivial state). Eq.10-77 Gable Rhodes, February 6th, 2012

Raising & Lowering Operators -Eigenvectors Once we have a ground state, repeated application of the raising operator will result in an infinite set of eigenvectors with distinct (non-degenerate) eigenvalues. And introducing an arbitrary starting point λ0. But for the special case of the ground state Must be zero on the right side (by our definition), so λ0 is exactly 0. Gable Rhodes, February 6th, 2012

Determine the Energy Levels Using the previous equation relating the Hamiltonian and aa†, we can relate energy to λ (n). And since we started at the ground state, we can relate the energy level to the eigenvectors If we use a normalization constant Where An can be found by direct integration at each step or by algebraic tricks (later) Gable Rhodes, February 6th, 2012

What Are These Operators Good For Anyway? We found Energy exactly and wavevectors in abstract form. What else can we do with them? What about expectation values? In chapter 5, problem 2, we were asked to find <V>. This required direct integration with the (explicitly known) wavefunction. Can this be done without knowing the wavefunction? Gable Rhodes, February 6th, 2012

Expectation value of Potential If we use the definition of a†, a and rewrite q and p operators in terms of a and a† we get Now these can be substituted into <V> Gable Rhodes, February 6th, 2012

Expectation value of Potential Of the four terms in the integral, we see that two of them vanish due to the orthogonality of the wavevectors. And the other two are known to us from the previous work. Gable Rhodes, February 6th, 2012

Expectation value of Potential This makes the result pretty straightforward And the this result agrees with problem 5.2, and the virial theorem But, we did not need to know the explicit form of the wavefunction. Gable Rhodes, February 6th, 2012

<q>, <p>, <q2>, <p2> Other key expectation values can be easily obtained. Gable Rhodes, February 6th, 2012

Variance and Uncertainty? The variance is therefore And this agrees with the uncertainty principle Gable Rhodes, February 6th, 2012

What is left then? The Normalization constant, which can also be determined algebraically. Square both sides and integrate After integration by parts and throwing out the boundary terms. So that gives us our wavefunction in terms of the ground state and raising operator Gable Rhodes, February 6th, 2012

Normalization Lets try ψ1. Next try ψ2. We can now write the general equation Gable Rhodes, February 6th, 2012

And what is ψ0? Starting with our condition for the ground state And using the definition of the operator We get a first order ODE Gable Rhodes, February 6th, 2012

And what is ψ0? The equation is separable and easily solved. After normalization we get Which is consistent with the solutions in chapter 5. Gable Rhodes, February 6th, 2012

Finding ψ1. Here we can use the raising operator to generate further solutions Substitute in the operator After rearranging, we get the desired result. Gable Rhodes, February 6th, 2012

Finding ψn. We find that continuing up the ladder is in fact a generating algorithm for the Hermite polynomials, and the general equation then identical to eq. 5.39 Gable Rhodes, February 6th, 2012

Matrix representation of a, a† We can show that the lowering operator relates neighboring wave vectors with the normalization factor. Adding the bra. This gives the matrix elements of a as shown. Gable Rhodes, February 6th, 2012

Matrix representation of a, a† As an example, lowering n=2, Gable Rhodes, February 6th, 2012

Matrix representation of a, a† The equivalent representation of the raising operator is derived from the expression Adding the bra. Gable Rhodes, February 6th, 2012

Matrix representation of a, a† With the lowering and raising operators in matrix form, we can then solve for the q and p operators in terms of a, a†. For position we get, Gable Rhodes, February 6th, 2012

Matrix representation of a, a† And the equivalent expression for momentum is, The complex constant insures that the anti-symmetric matrix is Hermitian. Gable Rhodes, February 6th, 2012