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equations:schroedinger_equation

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 equations:schroedinger_equation [2019/05/21 17:12]michael Latex footnotes don't work, use docuwiki syntax of (()) instead equations:schroedinger_equation [2020/04/10 11:44] (current)109.81.208.52 [Concrete] Both sides previous revision Previous revision 2020/04/10 11:44 [Concrete] 2020/04/10 11:40 [Concrete] 2019/05/21 17:12 michael Latex footnotes don't work, use docuwiki syntax of (()) instead2019/05/21 16:55 michael remove duplicated line in free particle example2019/05/21 16:53 michael minus sign and hbar missing from momentum operator2018/05/14 07:05 jakobadmin [Concrete] 2018/05/12 12:59 jakobadmin [Concrete] 2018/05/12 12:59 jakobadmin [Concrete] 2018/05/12 12:58 jakobadmin [Concrete] 2018/05/12 12:57 jakobadmin [Concrete] 2018/05/11 15:32 jakobadmin [Concrete] 2018/05/11 15:04 jakobadmin [Concrete] 2018/05/11 15:02 jakobadmin [Concrete] 2018/05/11 15:02 jakobadmin [Concrete] 2018/05/11 15:02 jakobadmin [Concrete] 2018/05/09 18:56 jakobadmin [Concrete] 2018/05/09 18:55 jakobadmin [Concrete] 2018/05/09 18:54 jakobadmin [Concrete] 2018/05/09 18:52 [Concrete] 2018/05/09 18:52 [Concrete] 2018/05/09 18:50 [Concrete] 2018/05/09 18:49 [Concrete] 2018/05/09 18:47 [Concrete] 2018/05/09 18:46 [Concrete] 2018/05/09 18:34 [Concrete] 2018/05/09 15:38 jakobadmin [Concrete] 2018/05/09 13:55 jakobadmin [Intuitive] 2018/05/09 13:54 jakobadmin [Concrete] Next revision Previous revision 2020/04/10 11:44 [Concrete] 2020/04/10 11:40 [Concrete] 2019/05/21 17:12 michael Latex footnotes don't work, use docuwiki syntax of (()) instead2019/05/21 16:55 michael remove duplicated line in free particle example2019/05/21 16:53 michael minus sign and hbar missing from momentum operator2018/05/14 07:05 jakobadmin [Concrete] 2018/05/12 12:59 jakobadmin [Concrete] 2018/05/12 12:59 jakobadmin [Concrete] 2018/05/12 12:58 jakobadmin [Concrete] 2018/05/12 12:57 jakobadmin [Concrete] 2018/05/11 15:32 jakobadmin [Concrete] 2018/05/11 15:04 jakobadmin [Concrete] 2018/05/11 15:02 jakobadmin [Concrete] 2018/05/11 15:02 jakobadmin [Concrete] 2018/05/11 15:02 jakobadmin [Concrete] 2018/05/09 18:56 jakobadmin [Concrete] 2018/05/09 18:55 jakobadmin [Concrete] 2018/05/09 18:54 jakobadmin [Concrete] 2018/05/09 18:52 [Concrete] 2018/05/09 18:52 [Concrete] 2018/05/09 18:50 [Concrete] 2018/05/09 18:49 [Concrete] 2018/05/09 18:47 [Concrete] 2018/05/09 18:46 [Concrete] 2018/05/09 18:34 [Concrete] 2018/05/09 15:38 jakobadmin [Concrete] 2018/05/09 13:55 jakobadmin [Intuitive] 2018/05/09 13:54 jakobadmin [Concrete] 2018/05/09 13:51 jakobadmin [Concrete] 2018/05/09 13:50 jakobadmin [Concrete] 2018/05/09 13:36 jakobadmin [Concrete] 2018/05/09 13:33 jakobadmin [Concrete] 2018/05/05 14:03 jakobadmin ↷ Links adapted because of a move operation2018/05/05 12:29 jakobadmin ↷ Links adapted because of a move operation2018/05/04 09:42 jakobadmin ↷ Links adapted because of a move operation2018/05/03 13:04 jakobadmin ↷ Links adapted because of a move operation2018/05/03 12:54 ↷ Links adapted because of a move operation2018/04/30 12:50 jakobadmin 2018/04/30 12:49 jakobadmin [Abstract] 2018/04/30 12:49 jakobadmin [Abstract] 2018/04/30 12:48 jakobadmin 2018/04/16 09:07 jakobadmin [Intuitive] 2018/04/12 16:15 bogumilvidovic ↷ Links adapted because of a move operation2018/04/09 15:31 tesmitekle [Abstract] 2018/04/09 15:31 tesmitekle [Abstract] 2018/04/09 15:29 tesmitekle 2018/04/09 11:01 tesmitekle [FAQ] 2018/04/08 16:13 jakobadmin ↷ Links adapted because of a move operation2018/03/28 10:06 jakobadmin 2018/03/28 10:05 jakobadmin 2018/03/28 08:58 jakobadmin [Why is it interesting?] Line 26: Line 26:  ​  ​ - In the Schrödinger equation, the ​wave-function​ $\color{firebrick}{\Psi(\vec{x},​t)}$ 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 ​wave-function​ $\color{firebrick}{\Psi(\vec{x},​t)}$ 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 ​ of the wave function. We act with the ​partial derivative​ $\color{darkturquoise}{\partial_t}$ 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 ​ of the wave function. We act with the ​partial derivative​ $\color{darkturquoise}{\partial_t}$ on our wave-function and the result: ${\partial_t}{ \Psi(\vec{x},​t)}$ denotes how the wave function changes as time passes on. Line 67: Line 67: $$\phi(t) = A e^{-Et/​\hbar}$$ $$\phi(t) = A e^{-Et/​\hbar}$$ - 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__