What is the expansion postulate

[Greg Bernhardt]

Definition/Summary

The Expansion Postulate is a fundamental postulate in the formalism of quantum mechanics which states that any wavefunction that describes a possible state of a quantum system can be written as a linear combination of the eigenfunctions of a linear hermitian operator.

Mathematically, this means that these eigenfunctions form a complete basis set for the Hilbert space.

Equations

For operators with a discrete eigenvalue spectrum:

$$\psi(x)=\sum_i C_i \phi_i(x)$$

where the $\phi_i$'s are the eigenfunctions.

The coefficients, $C_i$, are:

$$C_i = \int \phi_i^*(x) \psi(x) dx$$

For operators with a continuous eigenvalue spectrum:

$$\psi (x)=\int C_{x_o}\delta (x_o- x) dx_o$$

In Dirac notation:

$$|\psi>=|\psi_i><\psi|\phi_i>$$

Extended explanation

The expansion postulate is encountered frequently when doing quantum mechanics.

We have a quantum system in a general state described by the wavefunction $\psi (x)$. Assuming we know the potential the system is in, we can solve the time-independent Schrodinger equation for the energy eigenstates of the system, $\phi_i (x)$. The general state of the system can then be expressed as:

$$\psi(x)=\sum_i C_i \phi_i(x)$$

This is the expansion postulate in action. Furthermore, if we know the wavefunction describing the initial state, Fourier analysis can be used to find expressions for the coefficients, $C_i$. Specifically,

$$C_i = <\psi|\phi_i> = \int \phi_i^*(x) \psi(x) dx$$

Now, any measurement of an observable of the system, even if it is in a general state as above, will always return an eigenvalue of the operator corresponding to the observable. In our case above, any measurement of the energy of the system in state $\psi(x)$ will force the system to collapse into one of the energy eigenstates. The probability the the system will collapse into the ith energy state, i.e. return the ith energy eigenvalue upon measurement is given by $|C_i|^2$.

Let's consider a standard example- a particle in an infinite square well:

The eigenfunctions of the time independent Schrodinger equation, i.e. the eigenfunctions of the Hamiltonian operator, are:

$$\phi_n(x)=\sqrt{\frac{2}{a}}\sin(\frac{n\pi x}{a})$$

where $n=1,2,3,...$

Let's assume the particle is in some general state, $\psi(x)$. This state can be expressed as:

$$\psi (x)=\sqrt{\frac{2}{a}}\sum_{n=1}^{\infty}C_n \sin(\frac{n\pi x}{a})$$

If we knew the closed form for $\psi (x)$, we could find the coefficents:

$$C_n=\int\psi (x)\sin(\frac{n\pi x}{a}) dx$$

The expansion postulate applies to eigenfunctions of all hermitian operators, not just the Hamiltonian operator.

For instance, consider the eigenfunctions of a spin 1/2 particle (expressed as matrices):

$$|\uparrow>=\left(\begin{array}{c} 1\\ 0 \end{array}\right)$$

and

$$|\downarrow>=\left(\begin{array}{c}0\\1\end{array}\right)$$

Both of these are eigenfunctions of the z spin operator:

$$S_z = \hbar\left(\begin{array}{cc}1/2&0\\0&-1/2\end{array}\right)$$

Thus, any general spin state the particle can be in can be expressed as:

$$|\psi>=a|\uparrow>+b|\downarrow>$$

where $|a|^2$ and $|b|^2$ are the probabilities that a measurement of the particles spin will return spin up or spin down, respectively.

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[AaronS]

Thanks for the overview of expansion postulate