What is quantum mechanics?
-the description of the behaviour of matter and light in all its details and, in particular, of the happenings of an atomic scale
Waves or Particles?
- Newton thought that light was made up of particles
- later it was discovered that it behaved like a wave
- now we say that it is like neither (or both)
Quantum Behaviour of Atomic Objects
- electrons behave like light
- the quantum behaviour of atomic objects (electrons, protons, neutrons, photons etc.) is the same for all
- they are all ‘particle waves’ or whatever you want to call them
Quantum Mechanics Timeline
1901 - Planck: Blackbody radiation 1905 - Einstein: Photoelectric effect 1913 - Bohr: Atomic spectra 1922 - Compton: Scattering photons off electrons 1924 - Pauli: Exclusion principle 1925 - de Broglie: Matter waves 1927 - Heisenberg: Uncertainty principle 1927 - Born: Interpretation of the wave function
Double Slit Experiment - With Bullets
- bullets fired in all directions
- pass through slits
- hit a detector screen
- the probability when one slit is closed gives a peak directly behind the slit
- when both are open the pattern is the sum of the one from each slit individually
Double Slit Experiment - With Water Waves
-for each slit individually the same pattern for the bullets is produced
-but when both slits are open, an interference pattern is produced
I12 = I1 + I2 + 2√(I1I2) cos𝛿
Double Slit Experiment - With Electrons
- for each slit individually the result is the same as for the bullets
- when both slits are open an interference pattern is produced
- HOWEVER
- if the electrons are observed as they pass through the slit then the same result as for the bullets with both slits open is observed
Why is the curve smooth for bullets but not for electrons (unobserved)?
- they are not actually different
- but for macroscopic objects like bullets the interference pattern is so narrow that the curve appears smooth
probability amplitude
definition
-the probability of an event in an ideal experiment is given by the absolute value of a complex number ϕ:
P = |ϕ|² = ϕ ϕ*
-where ϕ* is the complex conjugate
Interference Present - Summary
-when an event can occur in several alternative ways, the probability amplitude for the event is the sum of the probability amplitudes for each way considered separately, there is interference:
P = | h1 + h2 + … |²
Interference Lost - Summary
-if an experiment is performed which is capable of determining whether one or another alternative is actually taken, the probability of the event is the sum of the probabilities for each alternative:
P = |h1|² + |h2|² + …
Double Slit - Actual Experiment
- two possible paths, same collection point
- one trajectory passes through a quantum dot (the dot serves as an electron weigh station , so knows if the electron has gone that way)
1. when the detector sensitivity is low, an interference pattern is seen
2. when the detector is sensitive enough to detect an electron passing, interference is lost
Classical Wave Equations
v² * ∂²y(x,t)/∂x² = ∂²y(x,t)/∂t²
-this is solved by:
y(x,t) = A sin(kx - ωt)
ω/k = v
Time Dependent Schrodinger Equation
-ħ²/2m * ∂²Ψ(x,t)/∂x² = iħ * ∂Ψ(x,t)/∂t
Solution to the Time Dependent Schrodinger Equation
-starting with the time dependent form, solve using separation of variables to obtain:
Ψ = Ψ(x) * e^(-iEt/ħ)
Time Independent Schrodinger Equation
-starting with the time dependent Schrodinger equation, sub in the solution:
Ψ = Ψ(x) * e^(-iEt/ħ)
-cancel out the exponential function on both sides to obtain the time independent form:
-ħ²/2m * ∂²Ψ/∂x² = E * Ψ(x)
Schrodinger Equation Modified for an External Potential
(-ħ²/2m ∂²/∂x² + U(x)) * Ψ(x) = E * Ψ(x)
- where -ħ²/2m ∂²/∂x² + U(x) is the Hamiltonian
- and -ħ²/2m ∂²/∂x² is kinetic energy
- and U(x) is the potential energy
Problems with the Schrodinger Equation
1) we cannot derive it from first principles as it itself is a postulate of quantum mechanics
2) it is a complex equation (contains i) but how can we measure a complex wave function?
Solution to the Problems with the Wave Equation
-instead of measuring a complex wave function we measure the probability density:
|Ψ(x,t)|²
Probability Density
-the probability of finding a particle between x and x+dx
-given by:
|Ψ(x,t)|²
Normalising Probability Density
-as the particle must be somewhere in space, integrating the probability density over all space must give 1:
∫ |Ψ(x,t)|² dx = 1
-where the integral is taken from -∞ to +∞
How to Normalise a Wave Function
-given an un-normalised wave function Ψ(x) :
Ψ(x) / √[ ∫ |Ψ(x)|² dx] = normalised wave function
Example - The Plane Wave
Ψ(x,t) = A*e^(i(kx-ωt)
-sub into the Schrodinger Equation to obtain:
ħ²k²/2m = ħω
-which is correct as both sides = E, so a plane wave satisfies the Schrodinger Equation
-probability density:
|Ψ(x,t)|² = A²
-a particle that can be found any where in space with equal probability i.e. a wave
Does an ordinary wave f(x) = Asin(kx-wt) solve the Schrodinger Equation?
NO
/2m ≥ (ħ/2L)²/2m > 0 -the particle constantly 'wiggles' with some kinetic energy, even at absolute zero, this is referred to as zero point motion
