# 4.3: Observable Quantities Must Be Eigenvalues of Quantum Mechanical Operators

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The Laplacian operator \(\nabla\) is called an operator because it does something to the function that follows: namely, it produces or generates the sum of the three second-derivatives of the function. Of course, this is not done automatically; you must do the work, or remember to use this operator properly in algebraic manipulations. Symbols for operators are often (although not always) denoted by a hat ^ over the symbol, unless the symbol is used exclusively for an operator, e.g. \(\nabla\) (del/nabla), or does not involve differentiation, e.g.\(r\) for position.

Recall, that we can identify the total energy operator, which is called the Hamiltonian operator, \(\hat{H}\), as consisting of the kinetic energy operator plus the potential energy operator.

\[\hat {H} = - \dfrac {\hbar ^2}{2m} \nabla ^2 + \hat {V} (x, y , z ) \label{3-22}\]

Using this notation, we write the Schrödinger Equation as

\[ \hat {H} | \psi (x , y , z ) \rangle = E | \psi ( x , y , z ) \rangle \label{3-23}\]

Equation \(\ref{3-23}\) says that the Hamiltonian operator operates on the wavefunction to produce the energy, which is a number, (a quantity of Joules), times the wavefunction. Such an equation, where the operator, operating on a function, produces a constant times the function, is called an eigenvalue equation. The function is called an eigenfunction, and the resulting numerical value is called the eigenvalue. Eigen here is the German word meaning self or own.

It is a general principle of Quantum Mechanics that there is an operator for every physical observable. A physical observable is anything that can be measured. If the wavefunction that describes a system is an eigenfunction of an operator, then the value of the associated observable is extracted from the eigenfunction by operating on the eigenfunction with the appropriate operator. The value of the observable for the system is the eigenvalue, and the system is said to be in an eigenstate. Equation \(\ref{3-23}\) states this principle mathematically for the case of energy as the observable.

Consider a general real-space operator \(A(x)\). When this operator acts on a general wavefunction \(\psi(x)\) the result is usually a wavefunction with a completely different shape. However, there are certain special wavefunctions which are such that when \(A\) acts on them the result is just a multiple of the original wavefunction. These special wavefunctions are called *eigenstates*, and the multiples are called *eigenvalues*. Thus, if

\[A | \psi_a(x) \rangle = a | \psi_a(x) \rangle \label{4.3.2}\]

where \(a\) is a complex number, then \(\psi_a\) is called an **eigenstate **of \(A\) corresponding to the eigenvalue \(a\).

Suppose that \(A\) is an operator corresponding to some physical dynamical variable. Consider a particle whose wavefunction is \(\psi_a\). The expectation of value \(A\) in this state is simply

\[ \begin{align} \langle A\rangle &= \int_{-\infty}^\infty \psi_a^{\ast} A \psi_a dx \\[4pt] &= a \int_{-\infty}^\infty \psi_a^{\ast} \psi_a dx \\[4pt] &= a \label{4.3.3} \end{align}\]

where use has been made of Equation \(\ref{4.3.2}\) and the normalization condition. Moreover,

\[ \begin{align} \langle A^2\rangle &= \int_{-\infty}^\infty \psi_a^{\ast} A^2 \psi_a dx \\[4pt] &= a \int_{-\infty}^\infty \psi_a^{\ast} A \psi_a dx \\[4pt] &= a^2 \int_{-\infty}^\infty \psi_a^{\ast} \psi_a dx \\[4pt] &= a^2, \label{ 4.3.4} \end{align} \]

so the variance of \(A\) is

\[ \begin{align} \sigma_A^{ 2} &= \langle A^2\rangle - \langle A\rangle^2 = a^2-a^2 \\[4pt] &= 0. \label{4.3.5} \end{align} \]

The fact that the variance is *zero* implies that every measurement of \(A\) is bound to yield the same result: namely, \(a\). Thus, the eigenstate \(\psi_a\) is a state which is associated with a *unique* value of the dynamical variable corresponding to \(A\). This unique value is simply the associated eigenvalue determined by Equation \(\ref{4.3.2}\).

## Expectation Values

We have seen that \(\vert\psi(x,t)\vert^{ 2}\) is the probability density of a measurement of a particle's displacement yielding the value \(x\) at time \(t\). Suppose that we made a large number of independent measurements of the displacement on an equally large number of identical quantum systems. In general, measurements made on different systems will yield different results. However, from the definition of probability, the mean of all these results is simply

\[ \langle x\rangle = \int_{-\infty}^{\infty} x \vert\psi\vert^{ 2} dx \label{ 4.3.5}\]

Here, \(\langle x\rangle\) is called the *expectation value* of \(x\). Similarly the expectation value of any function of \(x\) is

\[ \langle f(x)\rangle = \int_{-\infty}^{\infty} f(x) \vert\psi\vert^{ 2} dx.\label{ 4.3.6}\]

If an unormalized wavefunction is used, then Equation \(\ref{4.3.7}\) changes to

\[ \begin{align} \langle a \rangle &= \dfrac{\langle \psi | a |\psi \rangle}{\langle \psi | \psi \rangle} \\[4pt] &=\dfrac{ \displaystyle \int_{-\infty}^{\infty} \psi^* \hat{A} \psi dx}{ \displaystyle \int_{-\infty}^{\infty} \psi^* \psi dx} \label{4.3.8} \end{align}\]

The denominator is just the normalization requirement discussed earlier. In general, the results of the various different measurements of \(x\) will be scattered around the expectation value \(\langle x\rangle\). The degree of scatter is parameterized by the quantity

\[ \begin{align} \sigma^2_x &= \int_{-\infty}^{\infty} \left(x-\langle x\rangle \right)^2 |\psi|^{ 2} dx \\[4pt] &\equiv \langle x^2\rangle -\langle x\rangle^{2}, \label{4.3.9} \end{align} \]

which is known as the *variance* of \(x\). The square-root of this quantity, \(\sigma_x\), is called the *standard deviation* of \(x\). We generally expect the results of measurements of \(x\) to lie within a few standard deviations of the expectation value (Figure \(\PageIndex{1}\)).

## Expanding the Wavefunction

It is also possible to demonstrate that the eigenstates of an operator attributed to a observable form a *complete set* (*i.e.*, that any general wavefunction can be written as a linear combination of these eigenstates). However, the proof is quite difficult, and we shall not attempt it here.

In summary, given an operator \(\hat{A}\), any general wavefunction, \(\psi(x)\), can be written

\[\psi = \sum_{i}c_i \psi_i\label{4.3.9A}\]

where the \(c_i\) are complex weights, and the \(\psi(x)\) are the properly normalized (and mutually orthogonal) eigenstates of \(\hat{A}\): i.e.,

\[A \psi_i = a_i \psi_i \label{4.3.10}\]

where \(a_i\) is the eigenvalue corresponding to the eigenstate \(\psi_i\), and

\[\int_{-\infty}^\infty \psi_i^\ast \psi_j dx = \delta_{ij}. \label{4.3.11}\]

Here, \(\delta_{ij}\) is called the *Kronecker delta-function*, and takes the value unity when its two indices are equal, and zero otherwise. It follows from Equations \(\ref{4.3.8}\) and \(\ref{4.3.11}\) that

\[ c_i = \int_{-\infty}^\infty \psi_i^\ast \psi dx. \label{4.3.12}\]

Thus, the expansion coefficients in Equation \(\ref{4.3.12}\) are easily determined, given the wavefunction \(\psi\) and the eigenstates \(\psi_i\). Moreover, if \(\psi\) is a properly normalized wavefunction then Equations \(\ref{4.3.8}\) and \(\ref{4.3.11}\) yield

\[ \sum_i \vert c_i\vert^2 =1. \label{4.3.13}\]

## Contributors

Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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