PHYS2041 Quantum Mechanics

Nota:

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

1. De Broglie wavelength

   Nota:

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

      Nota:

      • 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

         Nota:

         • 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

            Nota:

            • \(E = h \mu \) \( \lambda = \frac{h}{p} \) \[ E  = \frac{hc}{\lambda} = pc \] \( \mu \) - period
      2. kinetic Energy

         Nota:

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

      Nota:

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

Nota:

• comes in discrete portions -Enger in light particles

Nota:

• how heated bodies radiate 

1. Rayleigh-Jeans intensty spectrum result

   Nota:

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

   Nota:

   • -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

      Nota:

      • 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

   Nota:

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

Nota:

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

1. Hydrogen atom spectrum
2. Bohr's Rule

   Nota:

   • 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)

Nota:

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

1. The Schrodinger Equation

   Nota:

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

      Nota:

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

      Nota:

      • \[ <x> = \int_{-\infty}^{+\infty} x |\Psi (x, t)|^2 dx \] 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

   Nota:

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

      Nota:

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

1. Energy

   Nota:

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

1. length scale

   Nota:

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

   Nota:

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