PHYS2041 Quantum Mechanics

Annotations:

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

1. De Broglie wavelength

   Annotations:

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

      Annotations:

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

            Annotations:

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

         Annotations:

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

      Annotations:

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

Annotations:

• comes in discrete portions -Enger in light particles

Annotations:

• how heated bodies radiate 

1. Rayleigh-Jeans intensty spectrum result

   Annotations:

   • $$I(\lambda ) = \frac{8 \pi}{ \lambda^4} k_{B} T$$
2. E.M. radiation

   Annotations:

   • -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
   1. classically

      Annotations:

      • 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
3. Planck’s (quantum) radiation law

   Annotations:

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

Annotations:

• emission spectrum of atoms consists of just few (discrete) narrow spectral lines at certain wavelengths

1. Hydrogen atom spectrum
2. Bohr's Rule

   Annotations:

   • 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)

Annotations:

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

1. The Schrodinger Equation

   Annotations:

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

      Annotations:

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

      Annotations:

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

   Annotations:

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

      Annotations:

      • $$\hat{p} = -ih \frac{d}{dx}$$

1. Energy

   Annotations:

   • $$E_n = \frac{h^2}{2m}(\frac{\pi}{a})^2n^2$$
2. wave function

1. length scale

   Annotations:

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

   Annotations:

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