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equations:schroedinger_equation [2019/05/21 17:12] michael Latex footnotes don't work, use docuwiki syntax of `((<footnote>))` instead |
equations:schroedinger_equation [2020/04/10 11:40] 109.81.208.52 [Concrete] |
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- | In the Schrödinger equation, the <color firebrick>wave-function</color> $\color{firebrick}{\Psi(\vec{x},t)}$ describes the state of the system. This means concretely that ${ \Psi(\vec{x},t)}$ encodes, for example, if a given system is excited or in its rest state. The wave-function is a [[basic_tools:complex_analysis|complex function]], which means that at a given point in space, say $ \vec{x}= (1,4,7)$, and time, say $t=9s$, the wave function is a complex number: ${\Psi(1,4,7,9)}= 5 + 3 i$. The wave-function at a given point in space and time yields a probability amplitude. By taking the absolute square of an probability amplitude we get the probability density and by integrating over some spatial volume we get the probability to the the particle within this volume. | + | In the Schrödinger equation, the <color firebrick>wave-function</color> $\color{firebrick}{\Psi(\vec{x},t)}$ describes the state of the system. This means concretely that ${ \Psi(\vec{x},t)}$ encodes, for example, if a given system is excited or in its rest state. The wave-function is a [[basic_tools:complex_analysis|complex function]], which means that at a given point in space, say $ \vec{x}= (1,4,7)$, and time, say $t=9s$, the wave function is a complex number: ${\Psi(1,4,7,9)}= 5 + 3 i$. The wave-function at a given point in space and time yields a probability amplitude. By taking the absolute square of an probability amplitude we get the probability density and by integrating over some spatial volume we get the probability of the particle being within this volume. |
The left-hand side of the Schrödinger equation denotes the <color darkturquoise></color> of the wave function. We act with the <color darkturquoise>partial derivative</color> $\color{darkturquoise}{\partial_t}$ on our wave-function and the result: ${\partial_t}{ \Psi(\vec{x},t)}$ denotes how the wave function changes as time passes on. | The left-hand side of the Schrödinger equation denotes the <color darkturquoise></color> of the wave function. We act with the <color darkturquoise>partial derivative</color> $\color{darkturquoise}{\partial_t}$ on our wave-function and the result: ${\partial_t}{ \Psi(\vec{x},t)}$ denotes how the wave function changes as time passes on. |