Difference between revisions of "Quantum Mechanics"
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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 | 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. | where $m$ is the mass of the particle and $\omega$ is the angular frequency of the oscilator. | ||
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The 1D harmonic oscilator has known eigenstate solutions | 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 | where the functions Hn are the physicists' [https://en.wikipedia.org/wiki/Hermite_polynomials Hermite polynomials].Time propagation of eigenstates is described with | ||
\[\psi(t, x) = \mathrm e ^ {-i\omega t} \psi(x)\] | \[\psi(t, x) = \mathrm e ^ {-i\omega t} \psi(x)\] | ||
Revision as of 17:45, 21 December 2017
Introduction
The quantum world is governed by the Schrödinger equation
\[{\displaystyle {\hat {H}}|\psi (t)\rangle =i\hbar {\frac {\partial }{\partial t}}|\psi (t)\rangle } \]
where $\hat H$ is the Hamiltonian, $|\psi (t)\rangle$ is the quantum state function and $\hbar$ is the reduced 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 Hn are the physicists' Hermite polynomials.Time propagation of eigenstates is described with
\[\psi(t, x) = \mathrm e ^ {-i\omega t} \psi(x)\]
