Quantum Mechanics - Revision history

APirnat: Blanked the page

2019-05-21T09:12:59Z

Blanked the page

Gkosec at 14:24, 24 February 2018

2018-02-24T14:24:56Z

Mkolman: /* Particle in a box */

2018-01-11T13:55:40Z

‎Particle in a box

Mkolman at 13:39, 11 January 2018

2018-01-11T13:39:27Z

Mkolman at 12:38, 11 January 2018

2018-01-11T12:38:27Z

Mkolman: /* Harmonic oscilator */

2017-12-21T15:45:30Z

‎Harmonic oscilator

Mkolman: /* Introduction */

2017-12-21T15:45:14Z

‎Introduction

Mkolman: /* Introduction */

2017-12-21T15:44:42Z

‎Introduction

Mkolman: Created page with "= Introduction = The quantum world is governed by the [https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Schrödinger equation] \[{\displaystyle {\hat {H}}|\psi (t)\ran..."

2017-12-15T10:47:48Z

Created page with "= Introduction = The quantum world is governed by the [https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Schrödinger equation] \[{\displaystyle {\hat {H}}|\psi (t)\ran..."

New page

= Introduction =
The quantum world is governed by the [https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Schrödinger equation]

\[{\displaystyle {\hat {H}}|\psi (t)\rangle =i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle } \]

where $\hat H$ is the [https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics) Hamiltonian], $|\psi (t)\rangle$ is the [https://en.wikipedia.org/wiki/Wave_function quantum state function] and $\hbar$ is the reduced [https://en.wikipedia.org/wiki/Planck_constant Planck constant].

The Hamiltonian consists of kinetic energy $\hat T$ and potential energy $\hat V$. As in classical mechanics, potential energy is a function of time and space, whereas the kinetic energy differs from the classical world and is calculated as

\[\hat T = - \frac{\hbar^2}{2m} \nabla^2 .\]

The final version of the single particle Schrödinger equation can be written as

\[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) \psi(t, \mathbf r) = i\hbar {\frac {\partial }{\partial t}}\psi(t, \mathbf r) \]

← Older revision		Revision as of 09:12, 21 May 2019
Line 1:		Line 1:
−	~~Solution procedure is still compiling ... so please wait for results :)~~

−	~~= Introduction =~~
−	~~The quantum world is governed by the [https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Schrödinger equation]~~
−
−	~~\[{\displaystyle {\hat {H}}\|\psi (t)\rangle =i\hbar {\frac {\partial }{\partial t}}\|\psi (t)\rangle } \]~~
−
−	where $\hat H$ is the [https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics) Hamiltonian], $\|\psi (t)\rangle$ is the [https://en.wikipedia.org/wiki/Wave_function quantum state function] and $\hbar$ is the reduced [https://en.wikipedia.org/wiki/Planck_constant Planck constant].
−
−	The Hamiltonian consists of kinetic energy $\hat T$ and potential energy $\hat V$. As in classical mechanics, potential energy is a function of time and space, whereas the kinetic energy differs from the classical world and is calculated as
−
−	~~\[\hat T = - \frac{\hbar^2}{2m} \nabla^2 .\]~~
−
−	~~The final version of the single particle Schrödinger equation can be written as~~
−
−
−	~~\[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) \psi(t, \mathbf r) = i\hbar {\frac {\partial }{\partial t}}\psi(t, \mathbf r) \]~~
−
−	~~Quantum state function is a complex function, so it is usually split into the real part and imaginary part~~
−
−	~~\[ u, v \in C(\mathbb R)\colon \psi = u + i v , \]~~
−
−	~~which for a real $V$ yields a system of two real equations~~
−
−	~~\[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) u(t, \mathbf r) = -\hbar {\frac {\partial }{\partial t}} v(t, \mathbf r) , \]~~
−	~~\[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) v(t, \mathbf r) = \hbar {\frac {\partial }{\partial t}} u(t, \mathbf r) , \]~~
−
−	~~which may be easier to handle.~~
−
−	~~= Harmonic oscilator =~~
−
−	By selecting the potential $V(t, \mathbf r)$ and the initial state $\psi(0, \mathbf r)$ we get a unique solution for time propagation of the quantum state function. Probably the most used and well known example is the quantum harmonic oscilator, where we select a quadratic potential
−
−	~~\[V(t, \mathbf r) = V(\mathbf r) = \frac{1}{2} m \omega^2 r^2 , \]~~
−
−	~~where $m$ is the mass of the particle and $\omega$ is the angular frequency of the oscilator.~~
−
−	~~The 1D harmonic oscilator has known eigenstate solutions~~
−
−	\[\psi _{n}(x)={\frac {1}{\sqrt {2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad n=0,1,2,\ldots .\]
−
−	~~where the functions $H_n$ are the physicists' [https://en.wikipedia.org/wiki/Hermite_polynomials Hermite polynomials]. Time propagation of eigenstates is described with~~
−
−	~~\[\psi_n(t, x) = \mathrm e ^ {-i (n+0.5) \omega t} \psi_n(x)\]~~
−
−	~~= Particle in a box =~~
−
−	~~A theoretical one dimensional potential~~
−
−	~~\[\displaystyle V(x)={\begin{cases}0,&0<x<L,\\\infty ,&{\text{otherwise,}}\end{cases}}\]~~
−
−	~~is known as an infinite potential well. Its time independent eigenfunctions are~~
−
−	~~\[\sqrt{\frac{2}{L}}\psi_n(x) = \sin\left(k_n x \right), \qquad n = 1,2,3,...\]~~
−
−	~~where $k_n = \frac{\pi n}{L}$. With a time dependency similar to Harmonic oscilator~~
−
−	~~\[\psi_n(t, x) = \mathrm e ^ {-i \omega_n t} \psi_n(x),\]~~
−
−	~~where $\omega_n$ and $k_n$ are connected through dispersion relation through energy $E_n$~~
−
−	~~\[{\displaystyle E_{n}=\hbar \omega _{n}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2mL^{2}}}={\frac {\hbar ^{2} k_n^2}{2m}}}.\]~~

← Older revision		Revision as of 14:24, 24 February 2018
Line 1:		Line 1:
		+	Solution procedure is still compiling ... so please wait for results :)
		+
	= Introduction =		= Introduction =
	The quantum world is governed by the [https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Schrödinger equation]		The quantum world is governed by the [https://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation Schrödinger equation]

← Older revision		Revision as of 13:55, 11 January 2018
Line 58:		Line 58:
	where $\omega_n$ and $k_n$ are connected through dispersion relation through energy $E_n$		where $\omega_n$ and $k_n$ are connected through dispersion relation through energy $E_n$

−	\[{\displaystyle E_{n}=\hbar \omega _{n}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2mL^{2}}}={\frac {\hbar ^{2} k_n^2}{2m}}}\]	+	\[{\displaystyle E_{n}=\hbar \omega _{n}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2mL^{2}}}={\frac {\hbar ^{2} k_n^2}{2m}}}.\]

← Older revision		Revision as of 13:39, 11 January 2018
Line 41:		Line 41:

	\[\psi_n(t, x) = \mathrm e ^ {-i (n+0.5) \omega t} \psi_n(x)\]		\[\psi_n(t, x) = \mathrm e ^ {-i (n+0.5) \omega t} \psi_n(x)\]
		+
		+	= Particle in a box =
		+
		+	A theoretical one dimensional potential
		+
		+	\[\displaystyle V(x)={\begin{cases}0,&0<x<L,\\\infty ,&{\text{otherwise,}}\end{cases}}\]
		+
		+	is known as an infinite potential well. Its time independent eigenfunctions are
		+
		+	\[\sqrt{\frac{2}{L}}\psi_n(x) = \sin\left(k_n x \right), \qquad n = 1,2,3,...\]
		+
		+	where $k_n = \frac{\pi n}{L}$. With a time dependency similar to Harmonic oscilator
		+
		+	\[\psi_n(t, x) = \mathrm e ^ {-i \omega_n t} \psi_n(x),\]
		+
		+	where $\omega_n$ and $k_n$ are connected through dispersion relation through energy $E_n$
		+
		+	\[{\displaystyle E_{n}=\hbar \omega _{n}={\frac {n^{2}\pi ^{2}\hbar ^{2}}{2mL^{2}}}={\frac {\hbar ^{2} k_n^2}{2m}}}\]

@@ Line 38: / Line 38: @@
 \[\psi _{n}(x)={\frac {1}{\sqrt {2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad n=0,1,2,\ldots .\]
-where the functions Hn are the physicists' [https://en.wikipedia.org/wiki/Hermite_polynomials Hermite polynomials].Time propagation of eigenstates is described with
+where the functions $H_n$ are the physicists' [https://en.wikipedia.org/wiki/Hermite_polynomials Hermite polynomials]. Time propagation of eigenstates is described with
-\[\psi(t, x) = \mathrm e ^ {-i\omega t} \psi(x)\]
+\[\psi_n(t, x) = \mathrm e ^ {-i (n+0.5)  \omega t} \psi_n(x)\]

@@ Line 30: / Line 30: @@
 By selecting the potential $V(t, \mathbf r)$ and the initial state $\psi(0, \mathbf r)$ we get a unique solution for time propagation of the quantum state function. Probably the most used and well known example is the quantum harmonic oscilator, where we select a quadratic potential
-\[V(t, \mathbf r) = V(\mathbf r) = \frac{1}{2} m \omega^2 r^2 \],
+\[V(t, \mathbf r) = V(\mathbf r) = \frac{1}{2} m \omega^2 r^2 , \]
 where $m$ is the mass of the particle and $\omega$ is the angular frequency of the oscilator.
@@ Line 36: / Line 36: @@
 The 1D harmonic oscilator has known eigenstate solutions
-\[\psi _{n}(x)={\frac {1}{\sqrt {2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad n=0,1,2,\ldots .\],
+\[\psi _{n}(x)={\frac {1}{\sqrt {2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad n=0,1,2,\ldots .\]
 where the functions Hn are the physicists' [https://en.wikipedia.org/wiki/Hermite_polynomials Hermite polynomials].Time propagation of eigenstates is described with
 \[\psi(t, x) = \mathrm e ^ {-i\omega t} \psi(x)\]

@@ Line 17: / Line 17: @@
 Quantum state function is a complex function, so it is usually split into the real part and imaginary part
-\[ u, v \in C(\mathbb R)\colon \psi = u + i v \],
+\[ u, v \in C(\mathbb R)\colon \psi = u + i v , \]
 which for a real $V$ yields a system of two real equations
-\[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) u(t, \mathbf r) = -\hbar {\frac {\partial }{\partial t}} v(t, \mathbf r) \],
+\[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) u(t, \mathbf r) = -\hbar {\frac {\partial }{\partial t}} v(t, \mathbf r) , \]
-\[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) v(t, \mathbf r) = \hbar {\frac {\partial }{\partial t}} u(t, \mathbf r) \],
+\[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) v(t, \mathbf r) = \hbar {\frac {\partial }{\partial t}} u(t, \mathbf r) , \]
 which may be easier to handle.

← Older revision		Revision as of 15:44, 21 December 2017
Line 14:		Line 14:

	\[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) \psi(t, \mathbf r) = i\hbar {\frac {\partial }{\partial t}}\psi(t, \mathbf r) \]		\[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) \psi(t, \mathbf r) = i\hbar {\frac {\partial }{\partial t}}\psi(t, \mathbf r) \]
		+
		+	Quantum state function is a complex function, so it is usually split into the real part and imaginary part
		+
		+	\[ u, v \in C(\mathbb R)\colon \psi = u + i v \],
		+
		+	which for a real $V$ yields a system of two real equations
		+
		+	\[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) u(t, \mathbf r) = -\hbar {\frac {\partial }{\partial t}} v(t, \mathbf r) \],
		+	\[\left(- \frac{\hbar^2}{2m} \nabla^2 + V(t, \mathbf r)\right) v(t, \mathbf r) = \hbar {\frac {\partial }{\partial t}} u(t, \mathbf r) \],
		+
		+	which may be easier to handle.
		+
		+	= Harmonic oscilator =
		+
		+	By selecting the potential $V(t, \mathbf r)$ and the initial state $\psi(0, \mathbf r)$ we get a unique solution for time propagation of the quantum state function. Probably the most used and well known example is the quantum harmonic oscilator, where we select a quadratic potential
		+
		+	\[V(t, \mathbf r) = V(\mathbf r) = \frac{1}{2} m \omega^2 r^2 \],
		+
		+	where $m$ is the mass of the particle and $\omega$ is the angular frequency of the oscilator.
		+
		+	The 1D harmonic oscilator has known eigenstate solutions
		+
		+	\[\psi _{n}(x)={\frac {1}{\sqrt {2^{n}\,n!}}}\cdot \left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac {m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt {\frac {m\omega }{\hbar }}}x\right),\qquad n=0,1,2,\ldots .\],
		+
		+	where the functions Hn are the physicists' [https://en.wikipedia.org/wiki/Hermite_polynomials Hermite polynomials].Time propagation of eigenstates is described with
		+
		+	\[\psi(t, x) = \mathrm e ^ {-i\omega t} \psi(x)\]