PHYS2041 Quantum Mechanics

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Queensland Certificate of Education Physics Mindmap am PHYS2041 Quantum Mechanics, erstellt von Lucy Lowe am 24/07/2017.
Lucy Lowe
Mindmap von Lucy Lowe, aktualisiert more than 1 year ago
Lucy Lowe
Erstellt von Lucy Lowe vor mehr als 7 Jahre
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Zusammenfassung der Ressource

PHYS2041 Quantum Mechanics
  1. Wave-particle duality

    Anmerkungen:

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

      Anmerkungen:

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

        Anmerkungen:

        • 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

          Anmerkungen:

          • 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

            Anmerkungen:

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

            Anmerkungen:

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

            Anmerkungen:

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

          Anmerkungen:

          • comes in discrete portions -Enger in light particles
          1. Black body radiation

            Anmerkungen:

            • how heated bodies radiate 
            1. Rayleigh-Jeans intensty spectrum result

              Anmerkungen:

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

                Anmerkungen:

                • -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

                  Anmerkungen:

                  • 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
                2. Planck’s (quantum) radiation law

                  Anmerkungen:

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

                    Anmerkungen:

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

                        Anmerkungen:

                        • 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)
                      2. The wave function

                        Anmerkungen:

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

                          Anmerkungen:

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

                            Anmerkungen:

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

                              Anmerkungen:

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

                              Anmerkungen:

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

                                  Anmerkungen:

                                  • \[ \hat{p} = -ih \frac{d}{dx} \]
                              2. infinite well
                                1. Energy

                                  Anmerkungen:

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

                                      Anmerkungen:

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

                                        Anmerkungen:

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

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