| Version 11 |
Version 10 |
| 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 |
| \[ |
\[ |
| \psi \mapsto -\nabla^2\psi+V(x)\psi, \quad\psi\in L^2(\Rset^n). |
\psi \mapsto -\nabla^2\psi+V(x)\psi, \quad\psi\in L^2(\Rset^n). |
| \] |
\] |
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| This can be also written as: |
This can be also written as: |
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| \[ |
\[ |
| \psi \mapsto [-\nabla^2 +V(x)]\psi, \quad\psi\in L^2(\Rset^n), |
\psi \mapsto [-\nabla^2 +V(x)]\psi, \quad\psi\in L^2(\Rset^n), |
| \] where $[-\nabla^2 +V(x)]$ is the {\em Schr\"odinger operator}, or {\em Hamiltonian operator}, \textbf{H}, |
\] where $[-\nabla^2 +V(x)]$ is the {\em Schr\"odinger operator}, or {\em Hamiltonian operator}, \textbf{H}. |
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| For stationary quantum systems such as electrons in `stable' atoms the {\em Schr\"odinger equation} |
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| takes the very simple form : |
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| \[ |
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| \textbf{H} \psi=E \psi |
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| \] , where $E$ stands for energy eigenvalues of the stationary quantum states. |
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| In quantum mechanics, 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 `Hamilton' operator, or the Hamiltonian, 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 quantum mechanics, 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 `Hamilton' operator, or the Hamiltonian, 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. |
