3.10: Continuous Eigenvalues

In the previous two sections, it was tacitly assumed that we were dealing with operators possessing discrete eigenvalues and square-integrable eigenstates. Unfortunately, some operators—most notably, \(x\) and \(p\)—possess eigenvalues that lie in a continuous range and non-square-integrable eigenstates (in fact, these two properties go hand in hand). Let us, therefore, investigate the eigenstates and eigenvalues of the displacement and momentum operators.

Let \(\psi_x(x,x')\) be the eigenstate of \(x\) corresponding to the eigenvalue \(x'\). It follows that

\[x\,\psi_x(x,x') = x'\,\psi_x(x,x') \nonumber \]

for all \(x\). Consider the Dirac delta-function \(\delta(x-x')\). We can write

\[x\,\delta(x-x') = x'\,\delta(x-x'), \nonumber \]

because \(\delta(x-x')\) is only non-zero infinitesimally close to \(x=x'\). Evidently, \(\psi_x(x,x')\) is proportional to \(\delta(x-x')\). Let us make the constant of proportionality unity, so that

\[\psi_x(x,x') = \delta(x-x'). \nonumber \]

It is easily demonstrated that

\[\int_{-\infty}^{\infty} \delta(x-x')\,\delta(x-x'')\,dx = \delta(x'-x''). \nonumber \]

Hence, \(\psi_x(x,x')\) satisfies the orthonormality condition

\[ \int_{-\infty}^\infty \psi_x^\ast(x,x')\,\psi_x(x,x'')\,dx = \delta(x'-x''). \label{e4.143} \]

This condition is analogous to the orthonormality condition \ref{e3.125} satisfied by square-integrable eigenstates. Now, by definition, \(\delta(x-x')\) satisfies

\[\int_{-\infty}^\infty f(x)\,\delta(x-x')\,dx = f(x'), \nonumber \]

where \(f(x)\) is a general function. We can thus write

\[ \psi(x) = \int_{-\infty}^\infty c(x')\,\psi_x(x,x')\,dx', \label{e4.144} \]

where \(c(x')=\psi(x')\), or

\[ c(x') = \int_{-\infty}^\infty \psi_x^\ast(x,x')\,\psi(x)\,dx. \label{e4.145} \]

In other words, we can expand a general wavefunction \(\psi(x)\) as a linear combination of the eigenstates, \(\psi_x(x,x')\), of the displacement operator. Equations \ref{e4.144} and \ref{e4.145} are analogous to Equations \ref{e3.123} and \ref{e3.126}, respectively, for square-integrable eigenstates. Finally, by analogy with the results in Section 1.9, the probability density of a measurement of \(x\) yielding the value \(x'\) is \(|c(x')|^2\), which is equivalent to the standard result \(|\psi(x')|^2\). Moreover, these probabilities are properly normalized provided \(\psi(x)\) is properly normalized [cf., Equation \ref{e3.127}]: that is,

\[\int_{-\infty}^\infty |c(x')|^2\,dx'= \int_{-\infty}^\infty |\psi(x')|^2\,dx' =1. \nonumber \]

Finally, if a measurement of \(x\) yields the value \(x'\) then the system is left in the corresponding displacement eigenstate, \(\psi_x(x,x')\), immediately after the measurement. That is, the wavefunction collapses to a “spike-function”, \(\delta(x-x')\), as discussed in Section [scoll].

Now, an eigenstate of the momentum operator \(p\equiv -{\rm i}\,\hbar\,\partial/\partial x\) corresponding to the eigenvalue \(p'\) satisfies

\[-{\rm i}\,\hbar\,\frac{\partial \psi_p(x,p')}{\partial x} = p'\,\psi_p(x,p'). \nonumber \]

It is evident that

\[ \psi_p(x,p') \propto \rm e^{+{\rm i}\,p'\,x/\hbar}. \label{e4.148} \]

We require \(\psi_p(x,p')\) to satisfy an analogous orthonormality condition to Equation \ref{e4.143}: that is,

\[\int_{-\infty}^\infty \psi_p^\ast(x,p')\,\psi_p(x,p'')\,dx = \delta(p'-p''). \nonumber \]

Thus, it follows from Equation \ref{e3.72} that the constant of proportionality in Equation \ref{e4.148} should be \((2\pi\,\hbar)^{-1/2}\): that is,

\[ \psi_p(x,p') =\frac{ \rm e^{+{\rm i}\,p'\,x/\hbar}}{(2\pi\,\hbar)^{1/2}}. \label{e4.148a} \]

Furthermore, according to Equations \ref{e3.64} and \ref{e3.65},

\[ \psi(x) = \int_{-\infty}^\infty c(p')\,\psi_p(x,p')\,dp', \label{e4.152} \]

where \(c(p') = \phi(p')\) [see Equation \ref{e3.65}], or

\[ c(p') = \int_{-\infty}^\infty \psi_p^\ast(x,p')\,\psi(x)\,dx. \label{e4.153} \]

In other words, we can expand a general wavefunction \(\psi(x)\) as a linear combination of the eigenstates, \(\psi_p(x,p')\), of the momentum operator. Equations \ref{e4.152} and \ref{e4.153} are again analogous to Equations \ref{e3.123} and \ref{e3.126}, respectively, for square-integrable eigenstates. Likewise, the probability density of a measurement of \(p\) yielding the result \(p'\) is \(|c(p')|^2\), which is equivalent to the standard result \(|\phi(p')|^2\). The probabilities are also properly normalized provided \(\psi(x)\) is properly normalized [cf., Equation \ref{e3.83}]: that is,

\[\int_{-\infty}^\infty |c(p')|^2\,dp'= \int_{-\infty}^{\infty} |\phi(p')|^2\,dp' = \int_{-\infty}^\infty |\psi(x')|^2\,dx' =1. \nonumber \]

Finally, if a mesurement of \(p\) yields the value \(p'\) then the system is left in the corresponding momentum eigenstate, \(\psi_p(x,p')\), immediately after the measurement.