why is there eigenfunctions in quantum mechanics

[physica]

I am in my final year of physics (hoping to get into a theoretical physics masters). I have been going though quantum mechanics and I haven't seen an explanation as to why there is eigenfunctions in quantum mechanics.

 

From what I understand an eigenfunction is a function that stays the same once it's be derived or integrated. The eigenvalue is a constant that comes from this process. Last year I used eigenvalues to find points of equilibrium. This was a fairly easy concept to grasp. In quantum mechanics is the use of eigenfunctions fundamental like for instance quantisation? Or is it mathematically useful like using e for a general solution to a second order differential equation?

[elfmotat]

An eigenvalue is some number, call it [math]\lambda[/math], that satisfies the following equation:

 

[math]\hat{O} \psi = \lambda \psi[/math]

 

where [math]\hat{O}[/math] is some operator. If [math]\psi[/math] is a function, we call it an eigenfunction. In quantum mechanics all observables are associated with a hermitian operator (position, momentum, energy, etc.). The function [math]\psi[/math] itself satisfies the Schrodinger equation. Eigenvalues of each observable operator represent the possible values you might get if you were to take a measurement. For example, if we have solved the Schrodinger equation for a particular scenario and we're interested in the spectrum of possible momentum values we might measure for an electron, we simply find all numbers [math]p[/math] which satisfy the equation:

 

[math]\hat{P} \psi = p \psi[/math]

Edited November 13, 2014 by elfmotat

[studiot]

 

From what I understand an eigenfunction is a function that stays the same once it's be derived or integrated.

..............

Or is it mathematically useful like using e for a general solution to a second order differential equation?

 

 

Yes it is mathematically useful in many situations, both classical and quantum.

 

The significance of your first loose definition is that if a function is the same once differentiated (to a constant multiplier) then the solution to a differential equation such as

 

y'' = -ky is an eigenfunction.

 

In other words y is an eigenfunction of this equation.

 

The more general a(x)y''+b(x)y'+c(x)y = -ky

 

Is known as the general eignfunction problem and the area of maths to look up is called Sturm-Liouville theory.

 

S-L theory leads on to adjoint and self adjoint operators as elfmotat has indicated.

 

An important property of eignfunctions is orthogonality which leads them to be linearly independent and forms the basis of useful series solutions.

 

Returning to my first differential equation, this has a familiar general solution

 

y = p*cos(k^0.5x) + q*cos(k^0.5x) where p and q are constants

 

This may be recognised as the standing wave equation for a stretched string in classical mechanics if the boundary conditions y=0 at x =0 and x=a are added.

 

Using these b. conditions we see that p = 0 and q*sin(k^0.5a) = 0

 

For non trivial solutions since p = 0, q cannot equal 0 and therefore sin(k^0.5a) = 0.

 

This happens for the eigenvalue equation sin(k^0.5a) = 0, giving the nodes of the standing wave.

 

A similar result can be obtained with complex solutions to the Schrodinger equation in QM

[physica]

Many thanks for your input. So looking at this what is the role of eigenvalues in determining which energy values in are possible in a bound system?

[elfmotat]

Many thanks for your input. So looking at this what is the role of eigenvalues in determining which energy values in are possible in a bound system?

 

Solve the Schrodinger equation for whatever potential you're considering to get [math]\psi (\mathbf{r},t)[/math]. The allowed energy values [math]E[/math] are those which satisfy the equation:

 

[math]\hat{H} \psi ( \mathbf{r},t)=E \psi ( \mathbf{r},t)[/math]

 

where [math]\hat{H}[/math] is the Hamiltonian operator.

Edited November 21, 2014 by elfmotat

[studiot]

 

Solve the Schrodinger equation for whatever potential you're considering to get . The allowed energy values are those which satisfy the equation:

Remember that the general solution will not in general be eigenfunctions, these arise when you apply the boundary conditions.

The quantum equivalent of my stretched string is the 'particle in a box'.

[physica]

From what I'm getting if is not an eigenfunction the function will not converge when x goes to zero. Is this why energy levels are quantised? They can only jump between different eigenvalues???

[swansont]

From what I'm getting if is not an eigenfunction the function will not converge when x goes to zero. Is this why energy levels are quantised? They can only jump between different eigenvalues???

 

Yes. the eigenvalues are the only allowed values of the system.