| Version 21 |
Version 20 |
| Let $V\colon \Rset^n \to \Rset$ be a real-valued function. |
Let $V\colon \Rset^n \to \Rset$ be a real-valued function. |
| The \emph{Schr\"odinger operator} \textbf{H} on the Hilbert space $L^2(\Rset^n)$ is given by the action |
The \emph{Schr\"odinger operator} \textbf{H} on the Hilbert space $L^2(\Rset^n)$ is given by the action |
| \[ |
\[ |
| \psi \mapsto -\nabla^2\psi+V(x)\psi, \quad\psi\in L^2(\Rset^n). |
\psi \mapsto -\nabla^2\psi+V(x)\psi, \quad\psi\in L^2(\Rset^n). |
| \] |
\] |
|
|
| This can be obviously re-written as: |
This can be obviously re-written as: |
|
|
| \[ |
\[ |
| \psi \mapsto [-\nabla^2 +V(x)]\psi, \quad\psi\in L^2(\Rset^n), |
\psi \mapsto [-\nabla^2 +V(x)]\psi, \quad\psi\in L^2(\Rset^n), |
| \] where $[-\nabla^2 +V(x)]$ is the {\em Schr\"odinger} operator, which is now |
\] where $[-\nabla^2 +V(x)]$ is the {\em Schr\"odinger} operator, which is now |
| called the \PMlinkname{Hamiltonian operator}{HamiltonianOperatorOfAQuantumSystem}, \textbf{H}. |
called the \PMlinkname{Hamiltonian operator}{HamiltonianOperatorOfAQuantumSystem}, \textbf{H}. |
|
|
| For stationary quantum systems such as electrons in `stable' atoms the {\em Schr\"odinger equation} |
For stationary quantum systems such as electrons in `stable' atoms the {\em Schr\"odinger equation} |
| takes the very simple form : |
takes the very simple form : |
| \[ |
\[ |
| \textbf{H} \psi=E \psi |
\textbf{H} \psi=E \psi |
| \] , where $E$ stands for energy eigenvalues of the stationary quantum states. Thus, in quantum mechanics of systems with finite degrees of freedom that are `stationary', the Schr\"odinger operator is used to calculate the (time-independent) energy states of a quantum system with potential energy $V(x)$. Schr\"odinger called this operator the \PMlinkname{`Hamilton' operator}{ HamiltonianOperatorOfAQuantumSystem}, or the |
\] , where $E$ stands for energy eigenvalues of the stationary quantum states. Thus, in quantum mechanics of systems with finite degrees of freedom that are `stationary', the Schr\"odinger operator is used to calculate the (time-independent) energy states of a quantum system with potential energy $V(x)$. Schr\"odinger called this operator the \PMlinkname{`Hamilton' operator}{HamiltonOperator}, or the \PMlinkname{Hamiltonian}{HamiltonOperator}, and the latter name is currently used in almost all of quantum physics publications, etc. The eigenvalues give the energy levels, and the wavefunctions are given by the eigenfunctions. |
| \PMlinkname{Hamiltonian}{ HamiltonianOperatorOfAQuantumSystem}, and the latter name is currently used in almost all of quantum physics publications, etc. The eigenvalues give the energy levels, and the wavefunctions are given by the eigenfunctions. |
|
| In the more general, non-stationary, or `dynamic' case, the Schr\"odinger equation takes the general form: |
In the more general, non-stationary, or `dynamic' case, the Schr\"odinger equation takes the general form: |
|
|
| \[ |
\[ |
| \textbf{H} \psi= (-i) \partial \psi / \partial t |
\textbf{H} \psi= (-i) \partial \psi / \partial t |
| \]. |
\]. |
