# 3.7: Wavefunctions Must Be Normalized

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Since wavefunctions can in general be complex functions, the physical significance cannot be found from the function itself because the \(\sqrt {-1}\) is not a property of the physical world. Rather, the physical significance is found in the product of the wavefunction and its complex conjugate, i.e. the absolute square of the wavefunction, which also is called the square of the modulus.

\[ \Psi^*(r , t ) \psi (r , t ) = {|\Psi (r , t)|}^2 \label {3.6.1}\]

where \(r\) is a vector (x, y, z) specifying a point in three-dimensional space. The square is used, rather than the modulus itself, just like the intensity of a light wave depends on the square of the electric field. Remember that the Born interpretation is that \(\Psi^*(r_i)\Psi(r_i)\, d\tau\) is the *probability *that the electron is in the volume dτ located at \(r_i\). The Born interpretation therefore calls the wavefunction the probability amplitude, the absolute square of the wavefunction is called the **probability density**, and the probability density times a volume element in three-dimensional space (\(d\tau\)) is the probability.

## Normalization of the Wavefunction

A probability is a real number between 0 and 1, inclusive. An outcome of a measurement which has a probability 0 is an impossible outcome, whereas an outcome which has a probability 1 is a certain outcome. According to Equation \(\ref{3.6.1}\), the probability of a measurement of \(x\) yielding a result between \(-\infty\) and \(+\infty\) is

\[P_{x \in -\infty:\infty}(t) = \int_{-\infty}^{\infty}\vert\psi(x,t)\vert^{ 2} dx. \label{3.6.2}\]

However, a measurement of \(x\) *must* yield a value between \(-\infty\) and \(+\infty\), since the particle has to be located somewhere. It follows that \(P_{x \in -\infty:\infty}(t) =1\), or

\[\int_{-\infty}^{\infty}\vert\psi(x,t)\vert^{ 2} dx = 1, \label{3.6.3}\]

which is generally known as the **normalization condition** for the wavefunction.

## Time Dependence to the Wavefunction

Now, it is important to demonstrate that if a wavefunction is initially normalized then it stays normalized as it evolves in time according to the time-dependent Schrödinger's equation. If this is not the case then the probability interpretation of the wavefunction is untenable, since it does not make sense for the probability that a measurement of \(x\) yields *any* possible outcome (which is, manifestly, unity) to change in time. Hence, we require that

\[\dfrac{d}{dt} \int_{-\infty}^{\infty}\vert \psi(x,t)\vert^{ 2} dx = 0 \label{3.6.10}\]

for wavefunctions satisfying the time-dependent Schrödinger's equation (this results from the time-dependent Schrödinger's equation and Equation \(\ref{3.6.3}\)). The above equation gives

\[\dfrac{d}{dt} \int_{-\infty}^{\infty}\psi^* \psi dx=\int_{-\infty}^{\infty} \left(\dfrac{\partial\psi^*}{\partial t} \psi+\psi^* \dfrac{\partial\psi}{\partial t}\right) dx=0. \label{3.6.11}\]

Now, multiplying Schrödinger's equation by \(\psi^{*}/({\rm i} \hbar)\), we obtain

\[\psi^* \dfrac{\partial\psi}{\partial t}= \dfrac{\rm{i}}{2m} \psi^* \dfrac{\partial^2 \psi}{\partial x^2} - \dfrac{\rm{i}}{\hbar} V \vert \psi \vert^2 \label{3.6.12}\]

The complex conjugate of this expression yields

\[\psi \dfrac{\partial\psi^*}{\partial t}= -\dfrac{\rm{i}}{2m} \psi \dfrac{\partial^2 \psi^*}{\partial x^2} + \dfrac{\rm{i}}{\hbar} V \vert \psi \vert^2 \label{3.6.13}\]

since

- \((A B)^* = A^* B^*\),
- \(A^{* *}=A\), and
- \(i^*= -i\).

Summing Equation \ref{3.6.12} and \ref{3.6.13} results in

\[\dfrac{\partial\psi^*}{\partial t} \psi + \psi^{*} \dfrac{\partial\psi}{\partial t} = \dfrac{\rm{i}}{2m} \left(\psi^* \dfrac{\partial^2 \psi}{\partial x^2} - \psi \dfrac{\partial^2\psi^*}{\partial x^2}\right)=\dfrac{\rm{i}}{2m} \dfrac{\partial}{\partial x} \left(\psi^* \dfrac{\partial \psi}{\partial x} - \psi \dfrac{\partial\psi^*}{\partial x}\right). \label{3.6.14}\]

Equations \(\ref{3.6.11}\) and \(\ref{3.6.14}\) can be combined to produce

\[\dfrac{d}{dt}\int_{-\infty}^{\infty} \vert\psi\vert^2 dx= \dfrac{\rm{i}}{2m} \left[\psi^*\dfrac{\partial\psi}{\partial x}- \psi\dfrac{\partial\psi^*}{\partial x} \right]_{-\infty}^{\infty} = 0. \label{3.6.15}\]

The above equation is satisfied provided the wavefunction converges

\[\lim_{\vert x\vert\rightarrow\infty} \vert\psi\vert = 0 \label{3.6.16}\]

However, this is a necessary condition for the integral on the left-hand side of Equation \(\ref{3.6.3}\) to converge. Hence, we conclude that all wavefunctions which are *square-integrable* [i.e., are such that the integral in Equation \ref{3.6.3} converges] have the property that if the normalization condition Equation \(\ref{3.6.3}\) is satisfied at one instant in time then it is satisfied at all subsequent times.

## Contributors

David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules")

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