theories:quantum_mechanics:canonical
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theories:quantum_mechanics:canonical [2018/05/09 13:27] jakobadmin [Concrete] |
theories:quantum_mechanics:canonical [2020/04/02 13:23] 188.102.49.88 Fix typo |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| + | {{ :theories:quantum_mechanics:wavesqm.png?nolink&400|}} | ||
| - | In quantum mechanics, we no longer describe the trajectories of individual particles but only talk about probabilities that certain events can happen. | + | In quantum mechanics, we no longer describe the trajectories of individual particles but only talk about probabilities that certain events can happen. In the canonical description of quantum mechanics, we calculate these probabilities using a wave description for the particles. |
| So instead of describing the path between some points $A$ and $B$, we ask instead: "What's the probability that a particle which started at $A$ ends up at $B$?". | So instead of describing the path between some points $A$ and $B$, we ask instead: "What's the probability that a particle which started at $A$ ends up at $B$?". | ||
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| $$ \int_R dV |\Psi(\vec x,t)|^2 .$$ | $$ \int_R dV |\Psi(\vec x,t)|^2 .$$ | ||
| + | We get the wave function that describes the system in question by solving the [[equations:schroedinger_equation|Schrödinger equation]]. The object in the Schrödinger equation that describes the system in question is the [[formalisms:hamiltonian_formalism|Hamiltonian]] and the [[basic_notions:boundary_conditions|boundary conditions]]. | ||
| -->A short introduction# | -->A short introduction# | ||
| Line 216: | Line 217: | ||
| + | ---- | ||
| **Examples** | **Examples** | ||
| Line 298: | Line 300: | ||
| * Modern Quantum Mechanics by Jun John Sakurai | * Modern Quantum Mechanics by Jun John Sakurai | ||
| * The Principles of Quantum Mechanics by Paul Dirac | * The Principles of Quantum Mechanics by Paul Dirac | ||
| - | * Foundations of Quantum Mechanics by Gregory Naber for students who prefer a more mathematical treatment. | ||
| * Nice free lecture notes can be found [[https://www.colorado.edu/physics/phys7270/phys7270_fa16/lecnotes.html|here]]. | * Nice free lecture notes can be found [[https://www.colorado.edu/physics/phys7270/phys7270_fa16/lecnotes.html|here]]. | ||
| + | * See also [[https://www.isaacbooks.org/files/Ch1_qm2b.pdf|A Cavendish Quantum Mechanics Primer]] by M. Warner, FRS & A. C. H. Cheung which is a very gentle introduction aimed at highschool students. | ||
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| **The best abstract quantum mechanics textbooks are** | **The best abstract quantum mechanics textbooks are** | ||
| + | * Foundations of Quantum Mechanics by Gregory Naber for students who prefer a more mathematical treatment. | ||
| * Quantum Theory: Concepts and Methods by Asher Peres | * Quantum Theory: Concepts and Methods by Asher Peres | ||
| * Quantum Mechanics by L. E. Ballentine | * Quantum Mechanics by L. E. Ballentine | ||
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| + | <tabbox Equations> | ||
| + | |||
| + | The [[equations:schroedinger_equation|Schrödinger equation]] | ||
| + | |||
| + | $$i \hbar \partial_t \Psi(x,t) = H \Psi (x,t) $$ | ||
| + | |||
| + | that describes the time-evolution of the wave function. The time-independent version for systems where the Hamiltonian is time-independent is given by | ||
| + | |||
| + | $$H \psi(x)= E\psi(x),$$ | ||
| + | where the complete wave function $\Psi$ is then given by | ||
| + | |||
| + | $${\Psi(x,t) = \phi(t) \psi(x) = e^{-Et/\hbar} \psi(x)}$$ | ||
| + | ---- | ||
| + | |||
| + | The standard Hamiltonian is | ||
| + | |||
| + | $$ H = - \frac{\hbar^2}{2m} \Delta^2 + \hat V \hat{=} \frac{\hat{p}^2}{2m} +\hat{V}. $$ | ||
| + | ---- | ||
| + | |||
| + | The [[equations:heisenberg_equation|Heisenberg equation]] | ||
| + | |||
| + | $$\frac{\mathrm{d}\hat F}{\mathrm{d}t} = -\frac{i}{\hbar}[\hat F,\hat H] + \frac{\partial \hat F}{\partial t},$$ | ||
| + | |||
| + | which described the time-evolution of operators and the related time dependence of an expectation value | ||
| + | |||
| + | $$ \langle \frac{\mathrm{d}\hat F}{\mathrm{d}t} \rangle = \langle -\frac{i}{\hbar}[\hat F,\hat H] \rangle + \langle \frac{\partial \hat F}{\partial t} \rangle .$$ | ||
| + | |||
| + | ---- | ||
| + | |||
| + | The [[advanced_notions:uncertainty_principle|uncertainty principle]] | ||
| + | |||
| + | $$ \sigma_x \sigma_p \geq \hbar/2,$$ | ||
| + | |||
| + | and the generalized version | ||
| + | |||
| + | $$ \sigma_A \sigma_B \geq \big | \frac{1}{2i} \langle [A,B] \rangle \big|^2 .$$ | ||
| + | |||
| + | |||
| + | ---- | ||
| + | |||
| + | The [[formulas:canonical_commutation_relations|canonical commutation relations]] | ||
| + | $$ [\hat{p},\hat{x}] = -i \hbar .$$ | ||
| | | ||
theories/quantum_mechanics/canonical.txt · Last modified: 2020/04/02 14:39 by 62.178.252.198
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