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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 11:44] (current)
109.81.208.52 [Concrete]
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 <tabbox Concrete> ​ <tabbox Concrete> ​
-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. 
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 $$ \phi(t) = A e^{-Et/​\hbar} $$ $$ \phi(t) = A e^{-Et/​\hbar} $$
-and the second equation is known as the stationary Schrödnger equation. This means that for all systems where the Hamiltonian does not explicitly depend on the time, we known immediately ​how the time-dependence of the total wave function $\Psi(x,t)$ looks like, namely: $\Psi(x,t) = \phi(t) \psi(x) = A e^{-Et/​\hbar} \psi(x)$. The only thing we then have to do is to solve the __stationary Schrödinger equation__+and the second equation is known as the stationary Schrödnger equation. This means that for all systems where the Hamiltonian does not explicitly depend on the time, we know immediately ​what the time-dependence of the total wave function $\Psi(x,t)$ looks like, namely: $\Psi(x,t) = \phi(t) \psi(x) = A e^{-Et/​\hbar} \psi(x)$. The only thing we then have to do is to solve the __stationary Schrödinger equation__
  
 \begin{equation} \begin{equation}
equations/schroedinger_equation.1558451551.txt.gz · Last modified: 2019/05/21 15:12 (external edit)