Schrödinger Equation: Difference between revisions
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where <math>\psi(\textbf{r})</math> satisfies the time-independent Schrödinger equation, | where <math>\psi(\textbf{r})</math> satisfies the time-independent Schrödinger equation, | ||
<math> -\frac{\hbar^2}{2m}\nabla^2\psi+V(\textbf{r} | <math> -\frac{\hbar^2}{2m}\nabla^2\psi+V(\textbf{r})\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 | 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 |
Revision as of 14:39, 7 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. As a simple example of this equation, let us consider 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 Hamiltonian for this system 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 \mathcal{H}=\frac{p^2}{2m}+V(\textbf{r},t).}
To obtain the corresponding Schrödinger equation, we make the replacements, 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{p}\rightarrow\frac{\hbar}{i}\nabla} and 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 H\rightarrow i\hbar\frac{\partial}{\partial t}} . This turns both sides of the above equation into operators, which, when applied to the wave function 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)} , yields the 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 -\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 the solution can be written in the form,
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)=e^{-iEt/\hbar}\psi(\textbf{r}),}
where 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})} satisfies 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 -\frac{\hbar^2}{2m}\nabla^2\psi+V(\textbf{r})\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.
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, }
while the time-independent 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 \mathcal{H}|\psi\rangle=E|\psi\rangle. }
The single-particle case discussed before may be obtained from these more general equations by projecting them into position space.