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Section 7.9: Exploring Eigenvalue Equations
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The time-independent Schrödinger equation is an energy-eigenvalue equation. What does this mean? Symbolically (this symbolic notation is called Dirac notation) the act of an operator acting on a state can be expressed by
A |α> = β |β>, (7.17)
where A is an operator, β is typically a number, |α> and |β> are two different state vectors. In general, therefore, the action of an operator on a state produces a different state. For special states, however, the final state is the original state and we have
A |a> = a |a>, (7.18)
where |a> is an eigenvector or eigenstate of the operator A with eigenvalue a. This equation is called an eigenvalue equation. To find energy eigenfunctions of the time-independent Schrödinger equation, we solve Hψ = Eψ and solve the boundary conditions.
We can demonstrate the idea of operators, eigenvalues, and eigenvectors by using a 2 × 2 matrix to represent the operator,
A = A11 A12
A21 A22 (7.19)
and a two-element column vector,
|a> = a1
to represent the state vector. An eigenstate of the operator A has the property that the application of A to the eigenvector results in a new vector has the same direction as the original eigenvector and a magnitude that is a constant times the eigenvector's original magnitude. In other words: eigenvectors of A are stretched, not rotated, when the operator A is applied to them.
Select a matrix to begin. Restart. Drag the red vector around in the animation. The result of the matrix (operator) acting on the column vector (the state vector) is shown as the green vector. To make things easier, you may set the red vector's length equal to 1.
What are the two eigenvectors and eigenvalues of each operator?