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Version 19 |
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\begin{definition}
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{\em Definition 0.1:}
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| The Hamiltonian operator \textbf{H} introduced in quantum mechanics by Schr\"odinger (and thus sometimes also called the |
The Hamiltonian operator \textbf{H} introduced in quantum mechanics by Schr\"odinger (and thus sometimes also called the |
| \emph{Schr\"odinger operator}) on the Hilbert space $L^2(\Rset^n)$ is given by the action: |
\emph{Schr\"odinger operator}) on the Hilbert space $L^2(\Rset^n)$ is given by the action: |
| \[ |
\[ |
| \psi \mapsto [-\nabla^2 +V(x)]\psi, \quad\psi\in L^2(\Rset^n), |
\psi \mapsto [-\nabla^2 +V(x)]\psi, \quad\psi\in L^2(\Rset^n), |
| \] |
\] |
| \end{definition} |
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| The operator defined above $[-\nabla^2 +V(x)]$ , for a potential function $V(x)$ specified as the real-valued function $V\colon \Rset^n \to \Rset$ is called the {\em Hamiltonian operator}, \textbf{H}, and only very rarely the {\em Schr\"odinger operator}. The energy conservation (quantum) law written with the operator \textbf{H} as the |
The operator defined above $[-\nabla^2 +V(x)]$ , for a potential function $V(x)$ specified as the real-valued function $V\colon \Rset^n \to \Rset$ is called the {\em Hamiltonian operator}, \textbf{H}, and only very rarely the {\em Schr\"odinger operator}. The energy conservation (quantum) law written with the operator \textbf{H} as the |
| Schr\"odinger equation is fundamental in quantum mechanics and is perhaps the most utilized, mathematical computation device in quantum mechanics of systems with a finite number of degrees of freedom. There is also, however, the alternative approach in the Heisenberg picture, or formulation, in which the observable and other operators are time-dependent whereas the state vectors $\psi$ are time-independent, which reverses the time dependences betwen operators and state vectors from the more popular Schr\"odinger formulation. Other formulations of quantum theories occur in |
Schr\"odinger equation is fundamental in quantum mechanics and is perhaps the most utilized, mathematical computation device in quantum mechanics of systems with a finite number of degrees of freedom. There is also, however, the alternative approach in the Heisenberg picture, or formulation, in which the observable and other operators are time-dependent whereas the state vectors $\psi$ are time-independent, which reverses the time dependences betwen operators and state vectors from the more popular Schr\"odinger formulation. Other formulations of quantum theories occur in |
| quantum field theories (QFT), such as QED (quantum electrodynamics) and QCD (quantum chromodynamics). |
quantum field theories (QFT), such as QED (quantum electrodynamics) and QCD (quantum chromodynamics). |
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| Although the two formulations, or pictures, are unitarily (or mathematically) equivalent, however, sometimes the claim is made that the Heisenberg picture is ``more natural and fundamental than the Schr\"odinger'' formulation because the Lorentz invariance from General Relativity is also encountered in the Heisenberg picture, |
Although the two formulations, or pictures, are unitarily (or mathematically) equivalent, however, sometimes the claim is made that the Heisenberg picture is ``more natural and fundamental than the Schr\"odinger'' formulation because the Lorentz invariance from General Relativity is also encountered in the Heisenberg picture, |
| and also because there is a `correspondence' between the commutator of an observable operator with the Hamiltonian operator, and the Poisson bracket formulation of classical mechanics. If the state vector $\psi$, or |
and also because there is a `correspondence' between the commutator of an observable operator with the Hamiltonian operator, and the Poisson bracket formulation of classical mechanics. If the state vector $\psi$, or |
| $\left| \psi \right\rangle$ does not change with time as in the Heisenberg picture, then the `equation of motion' |
$\left| \psi \right\rangle$ does not change with time as in the Heisenberg picture, then the `equation of motion' |
| of a (quantum) observable operator is : |
of a (quantum) observable operator is : |
|
|
| \[ |
\[ |
| \frac{d}{dt} A_{quantum} = (i\hbar)^{-1}[A,H] + \left(\frac {\partial A}{\partial t}\right)_{classical} |
\frac{d}{dt} A_{quantum} = (i\hbar)^{-1}[A,H] + \left(\frac {\partial A}{\partial t}\right)_{classical} |
| \] |
\] |
