Schrödinger Equation: Difference between revisions
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<math> -\frac{\hbar^2}{2m}\nabla^2\psi+V(\textbf{r},t)\psi=E\psi. </math> | <math> -\frac{\hbar^2}{2m}\nabla^2\psi+V(\textbf{r},t)\psi=E\psi. </math> | ||
As stated before, the wave function is a measure of the probability that a particle is in a given state; in fact, the probability density of finding a particle at a position <math>\textbf{r}</math> at a given time <math>t</math> is <math>|\Psi(\textbf{r},t)|^2</math>, or <math>|\psi(\textbf{r})|^2</math> for the time-independent case. Because of this, the wave function must be normalized such that | |||
<math>\int d^3\textbf{r}\,|\Psi(\textbf{r},t)|^2 = 1,</math> | |||
or | |||
<math>\int d^3\textbf{r}\,|\psi(\textbf{r})|^2 = 1</math> | |||
in the time-independent case. We will also find that the Schrödinger equation respects conservation of probability, as expected. | |||
<b>Chapter Contents</b> | <b>Chapter Contents</b> | ||
Revision as of 12:17, 6 March 2013
In this chapter, we introduce the Schrödinger equation, the most fundamental equation in quantum mechanics. Given a Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} , this equation describes how the wave function evolves in time. In Dirac bra-ket notation, the (time-dependent) Schrödinger equation is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle=\mathcal{H}|\Psi(t)\rangle. }
If the Hamiltonian does not depend on time, then the wave function can be written as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi(t)\rangle=e^{-iEt/\hbar}|\psi\rangle, }
and we obtain the time-independent Schrödinger equation,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}|\psi\rangle=E|\psi\rangle. }
For a single particle in a potential Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(\textbf{r},t)} , the Schrödinger equation becomes, when projected onto position space,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{2m}\nabla^2\Psi+V(\textbf{r},t)\Psi=i\hbar\frac{\partial\Psi}{\partial t}. }
If the potential does not depend on time, then we obtain the time-independent form of the equation,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{2m}\nabla^2\psi+V(\textbf{r},t)\psi=E\psi. }
As stated before, the wave function is a measure of the probability that a particle is in a given state; in fact, the probability density of finding a particle at a position Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textbf{r}} at a given time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi(\textbf{r},t)|^2} , or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\psi(\textbf{r})|^2} for the time-independent case. Because of this, the wave function must be normalized such that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int d^3\textbf{r}\,|\Psi(\textbf{r},t)|^2 = 1,}
or
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int d^3\textbf{r}\,|\psi(\textbf{r})|^2 = 1}
in the time-independent case. We will also find that the Schrödinger equation respects conservation of probability, as expected.
Chapter Contents