# PHYS2041 Quantum Mechanics

Mind Map by Lucy Lowe, updated more than 1 year ago
 Created by Lucy Lowe almost 3 years ago
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### Description

Physics Mind Map on PHYS2041 Quantum Mechanics, created by Lucy Lowe on 07/24/2017.

## Resource summary

PHYS2041 Quantum Mechanics
1 Wave-particle duality

Annotations:

• every object has wave-like and particle-like properties (microscopic objects 'are’ particles and waves at the same time)
1.1 De Broglie wavelength

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• De Broglie wavelength $\lambda = \frac{h}{p}$ h = 6.24  x10-34 Js
1.1.1 non-relativistic particles

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• Momentum $p = mv$ $$m$$ -mass (kg) $$v = |v|$$ -speed $$h$$ - plank's constant $$6.62607004\times10-34 Js$$ wavelength $\lambda = \frac{h}{mv}$
1.1.1.1 particles of light

Annotations:

• photons = quanta of E.M radiation $p = hk = h \omega/c \rightarrow \lambda = \frac{h}{p} = \frac{2 \pi h}{p} = \frac{2 \pi h}{\omega} =Tc$   $$\lambda$$ - wavelength$$T$$ -oscillation period $$\omega$$ - frequency$$k = 2 \pi / \lambda$$ - wave-number
1.1.1.1.1 Energy of photon

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• $$E = h \mu$$ $$\lambda = \frac{h}{p}$$ $E = \frac{hc}{\lambda} = pc$ $$\mu$$ - period
1.1.1.2 kinetic Energy

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• $\frac{1}{2} mv^2 = \frac{1}{2} pv = \frac{p^2}{2m}$
1.1.2 momentum >= 0

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• Energy is never zero Always ground amount of energy p =mv = kg m/s
2 quantised

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• comes in discrete portions -Enger in light particles

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3.1 Rayleigh-Jeans intensty spectrum result

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• $I(\lambda ) = \frac{8 \pi}{ \lambda^4} k_{B} T$

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• -Field that permeates all space Max Planck (1900): Energy of E.M. radiation isquantised (comes in discrete portions): $E = nh \omega$$$n = 0,1,2,3,...$$ -  number of excitation quantah - planks constant$$\omega$$ - frequency
3.2.1 classically

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• Each standing wave or oscillator mode has two degrees of freedom classically, and should have an average thermal energy . $k_{B} T$ (classically) ultraviolet  catastrophe

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• $I(\lambda ) = \frac{8 \pi hc}{ \lambda^{5} \left(e^{\frac{hc}{ \lambda k_{B} T}} -1\right)}$
4 Photo-electric effect
5 Atomic spectra

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• emission spectrum of atoms consists of just few (discrete) narrow spectral lines at certain wavelengths
5.1 Hydrogen atom spectrum
5.2 Bohr's Rule

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• 2π x (electron mass) x (electron orbital speed) x (orbit radius) = (any integer) x h
• The energy lost by the electron is carried away by a photon: photon energy = (e’s energy in larger orbit) - (e’s energy in smaller orbit)
6 The wave function

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• Can only describe quantum systems when closed system (pure states). Open systems are described by density matrix.
6.1 The Schrodinger Equation

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• $ih \frac{ \Psi}{dt} = -\frac{h^2}{2m} \frac{d^2 \Psi}{dx^2} + V(x,t) \Psi$
6.1.1 The particle must be somewhere

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• $\int_{- \infty}^{\infty} |\Psi( x,t)|^2 dx = 1$
6.2 Normalisation
6.2.1 probabilty density

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• $&lt;x&gt; = \int_{-\infty}^{+\infty} x |\Psi (x, t)|^2 dx$ expectation value of x^2 $&lt;x^2&gt; = \int_{-\infty}^{+\infty} x^2 |\Psi (x, t)|^2 dx$
• mean variance of particle position, standard deviation. $\alpha_{x} = \sqrt{&lt;(\Delta x)^2&gt;} = \sqrt{ &lt;x^2&gt; - &lt;x&gt;^2}$
6.3 Expectation or mean values

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• $\langle O \rangle = \int dx \psi*O(x,p) \psi$
6.4 coordinate representation
6.4.1 momentum operator

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• $\hat{p} = -ih \frac{d}{dx}$
7 infinite well
7.1 Energy

Annotations:

• $E_n = \frac{h^2}{2m}(\frac{\pi}{a})^2n^2$
7.2 wave function
8 harmonic oscillator
8.1 length scale

Annotations:

• $l_{ho} = \sqrt{\ hbar /m \omega}$
8.2 Properties of raising and lowering operators

Annotations:

• $\hat{a}_+ \psi_n = \sqrt{n+1}\psi_{n+1}$ $\hat{a}_- \psi_n = \sqrt{n}\psi_{n-1}$

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