2 Getting to Schrödinger’s wave equation Works for light (photons), why doesn’t it work for electrons?We found that solutions to this equation areorwith the constraintwhich can be writtenMultiplying by ħ we getwhich is just E=pcBut we know that E=pc only works for massless particles so this equation can’t work for electrons.
3 Getting to Schrödinger’s wave equation doesn’t work for electrons. What does?orNote that each derivative of x gives us a k (momentum) while each derivative of t gives us an w (energy).Equal numbers of derivatives result in E=pcFor massive particles we need kinetic energySo we need two derivatives of x for p2 but only one derivative of t for K.If we add in potential energy as well we get the Schrödinger equation…
4 = + The Schrödinger equation Kinetic energy Potential energy The Schrodinger equation for a matter wave in one dimension (x,t):Kinetic energyPotential energyTotal energy+=This is the time dependent Schrödinger equation (TDSE) (discussed in 7.11) and is also the most general form.This potential energy is a function of x and t. It gives the potential energy of the particle for any x and t. It is not intrinsic to the particles but something from the problem at hand.
5 Time independent Schrödinger equation In most physics situations (like hydrogen atom) the potential function U does not change in time so can write U(x,t) = U(x).In this case, we can separate Y(x,t) into y(x)f(t):We will then use the time independent Schrödinger equation (TISE) for the x-component of the wavefunction (lower case psi):
6 Given a potential energy function U(x), where would a particle naturally want to be? Where U(x) is highestWhere U(x) is lowestWhere U(x) < kinetic energyWhere U(x) > kinetic energyDoes not depend on V(x)Particles want to go to position of lowest potential energy, like a ball going downhill.U(x)x
7 The infinite square well (particle in a box) The potential energy function isx < 0: U(x) ≈ ∞x > a: U(x) ≈ ∞0 < x < a: U(x) = 0Potential EnergyxaThis is called the infinite square well (referring to the potential energy graph) or particle in a box (since the particle is trapped inside a 1D box of length a.We are interested in the region 0 < x < a where U(x) = 0 soBecomes (for the states in the box)
8 Guess a solution toHow about y(x) = Acos(kx)?which givesorThe total energy E is completely kinetic energy(because we set the potential energy U=0)
9 Infinite square well solution The functional form of the solution isa∞V(x)Now we apply the boundary conditionsorLHS: x = 0:12We also knowso that
10 For an infinite square well, what are the possible values for E? a∞V(x)Any value (E is not quantized)Putting into the TISEgivesso(just kinetic energy)Putting in the k quantization condition gives
11 Infinite square well (particle in a box) solution After applying boundary conditions we foundandwhich gives us an energy ofWhat is the lowest energy possible?Something else
12 Infinite square well (particle in a box) solution After applying boundary conditions we foundandwhich gives us an energy ofxU=0aE14E19E116E1n=4n=3n=2n=1EnergyThings to notice:Energies are quantized.Minimum energy E1 is not zero. This is a general principle of QM.
13 Infinite square well (particle in a box) solution After applying boundary conditions we foundandwhich gives us an energy ofThings to notice:xV=0aE14E19E116E1n=4n=3n=2n=1EnergyEnergies are quantized.Minimum energy E1 is not zero.Consistent with uncertainty principle. x is between 0 and a so Dx~a/2. Since DxDp≥ħ/2, must be uncertainty in p. But if E=0 then p=0 so Dp=0, violating the uncertainty principle.When a is large, energy levels get closer so energy becomes more like continuum (like classical result).
14 A grain of sandSuzy gently places a tiny grain of sand at the bottom of a very narrow and deep well. She says: “Because of the laws of QM, this grain of sand have a finite energy so it must be floating off the ground”. Liz says: “That is ridiculous – of course it is not levitating off the ground. Therefore sand must not be quantum.”. Who is correct?xU=0aE14E19E116E1n=4n=3n=2n=1EnergySuzyLizNeither – sand is “quantum” but “finite energy” does not mean that the sand is levitating, which would mean U is larger.