8.6.1: Hermitian Operators
- Page ID
- 456215
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- Recognize that all experimental obervables are obtained by Hermitian operators
Consideration of the quantum mechanical description of the particle-in-a-box exposed two important properties of quantum mechanical systems. We saw that the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position and momentum of the particle could not be determined exactly. We now examine the generality of these insights by stating and proving some fundamental theorems. These theorems use the Hermitian property of quantum mechanical operators that correspond to observables, which is discuss first.
Hermitian Operators
Since the eigenvalues of a quantum mechanical operator correspond to measurable quantities, the eigenvalues must be real, and consequently a quantum mechanical operator must be Hermitian. To prove this, we start with the premises that \(ψ\) and \(φ\) are functions, \(\int d\tau\) represents integration over all coordinates, and the operator \(\hat {A}\) is Hermitian by definition if
\[ \int \psi ^* \hat {A} \psi \,d\tau = \int (\hat {A} ^* \psi ^* ) \psi \,d\tau \label {4-37} \]
This equation means that the complex conjugate of \(\hat {A}\) can operate on \(ψ^*\) to produce the same result after integration as \(\hat {A}\) operating on \(φ\), followed by integration. To prove that a quantum mechanical operator \(\hat {A}\) is Hermitian, consider the eigenvalue equation and its complex conjugate.
\[\hat {A} \psi = a \psi \label {4-38} \]
\[\hat {A}^* \psi ^* = a^* \psi ^* = a \psi ^* \label {4-39} \]
Note that \(a^* = a\) because the eigenvalue is real. Multiply Equation \(\ref{4-38}\) and \(\ref{4-39}\) from the left by \(ψ^*\) and \(ψ\), respectively, and integrate over the full range of all the coordinates. Note that \(ψ\) is normalized. The results are
\[ \int \psi ^* \hat {A} \psi \,d\tau = a \int \psi ^* \psi \,d\tau = a \label {4-40} \]
\[ \int \psi \hat {A}^* \psi ^* \,d \tau = a \int \psi \psi ^* \,d\tau = a \label {4-41} \]
Since both integrals equal \(a\), they must be equivalent.
\[ \int \psi ^* \hat {A} \psi \,d\tau = \int \psi \hat {A}^* \psi ^* \,d\tau \label {4-42} \]
The operator acting on the function,
\[\hat {A}^* \int \psi ^* \hat {A} \psi \,d\tau = \int \psi \hat {A} ^* \psi ^* \,d\tau_* \nonumber \]
produces a new function. Since functions commute, Equation \(\ref{4-42}\) can be rewritten as
\[ \int \psi ^* \hat {A} \psi d\tau = \int (\hat {A}^*\psi ^*) \psi d\tau \label{4-43} \]
This equality means that \(\hat {A}\) is Hermitian.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)