Introduction Flashcards
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
Principle of Linear Superposition
- the Schrodinger equation is a linear differential equation therefore:
- if Ψ1(x,t) and Ψ2(x,t) are solutions of the Schrodinger equation, then c1*Ψ1(x,t) + c2Ψ2(x,t) is also a solution
- this can be shown simply by subbing into the schrodinger equation
Schrodinger’s Cat
Description
- cat is sealed in a box with a flask of poison and a radioactive source
- if an internal monitor detects radioactivity (e.g. single atom decay) the flask is shattered releasing the poison
Schrodinger’s Cat
Copenhagen Interpretation
- after a while the cat is simultaneously dead AND alive
- when the box is opened and the cat is observed the wave function collapses into one state or the other i.e. the cat is dead OR alive
Schrodinger’s Cat
Many Worlds Interpretation
- every event is a branch point
- the cat is both alive and dead regardless of observation
- BUT the ‘alive’ and ‘dead’ cats are in different branches of the universe that are equally real but cannot interact with each other
Copenhagen Interpretation
-a system stops being a superposition of states and becomes either one or the other when an observation takes place
What is uncertainty?
- measuring a quantity n times we get may get different measurements
- we can calculate a mean and a standard deviation ΔQ
- if ΔQ=0 we always measure the same values of Q
- if ΔQ≠0 results fluctuate around some value and we cannot be certain of a measurement
Uncertainty
Double Slit
- in the double slit experiment we can measure two things:
1) momentum of electrons, from trace on screen
2) position of electrons exiting the slit - if the width of the slit is large then Δpx is small as the electrons are less spread out on the screen
- if the width of the slit is small then Δpx is larger as the electrons are diffracted more and spread out on the screen
Heisenberg’s Uncertainty Principle
Δx Δpx ≥ ħ/2
-where px is momentum in the x direction
What does Heisenberg’s uncertainty principle tell us?
- because of a particles wave nature, it is theoretically impossible to measure both the particles position and its momentum along the same axis
- measurement always perturbs the system, usually this has non observable consequences for macroscopic object but, at an atomic level this is very important
Heisenberg Uncertainty Principle and Gaussian Wave Packets
-for most states the uncertainty principle is strictly greater than
-but there are special states for which the equals sign hold
-one such instance is the Gaussian wave packet:
Ψ(x) = C*e^(-(x/2e)²)
Heisenberg Uncertainty Principle and Zero Point Motion
-from the uncertainty principle we can show that if a particle is enclosed within a box, it can never be at rest:
-consider a box of width L
-since the particle is definitely in the box:
Δx ≤ L
-rearrange the uncertainty principle:
Δx Δp ≥ ħ/2
Δp ≥ ħ/2Δx
-sub in Δx ≤ L
Δp ≥ ħ/2Δx ≥ ħ/2L
-calculate average (to account for fluctuations) energy:
= <p>/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</p>
Heisenberg Uncertainty Principle in Three Dimensions
Δx Δpx ≥ ħ/2
Δy Δpy ≥ ħ/2
Δz Δpz ≥ ħ/2
Heisenberg Uncertainty Principle - Energy and Time
ΔE Δt ≥ ħ/2
What is a Fourier Transform used for?
- to analyse more complicated waves we can decompose them into harmonics
- applying the Fourier transform to a function gives the amplitude of different wave vectors each corresponding to a harmonic that makes up the function
Fourier Transform Equation
A(k) = ∫ dx f(x) e^(-ikx)
- where the integral is taken from -∞ to +∞
- and f(x) is the function of the wave of interest
Fourier Transform Inverse Equation
f(x) = ∫ dk/2π A(k) e^(ikx)
-where the integral is taken from -∞ to +∞
What do particles look like as waves?
- a particle can be visualised as a packet of waves created by adding together a continuum of harmonic waves
- this creates a perfectly localised wave packet or ‘particle’
