General Concepts
Initial condition: the exact condition at the start of the experiment. Essentially conditions at t0.
Information never disappears: While this concept has come into question, see Stephen Hawking, the general idea is that all the information of the system remains in tact. That we can understand how a system might have been in the past given a) current conditions and b) governing laws. Susskind defines information as the distinguishing aspect between things(https://www.youtube.com/watch?v=VvOZd_tbZ-w).
Wave: disturbance travelling through a medium. Electricity is a magnetic disturbance travelling through a medium. Water wave is caused by disturbance of the pebble dropped travelling through the medium in which it is dropped, water.
Symbols
Formula
Hamilton’s Equations
Maxwell Equations
Einstein Field Equations
Schrodinger Equation
Classical Mechanics
Hamilton’s Equations
These equations are everywhere and govern pretty much everything. The framework for all of physics.
Maxwell Equations
Electromagnetism and such.
Einstein Field Equations
Equations that explore the very basis of space and time, focus on gravity.
Michelson and Morley’s Luminiferous Ether Experiment
Quantum Mechanics
General Concepts
A device that measures is also considered to have prepared a system in a given state.
Unit vectors in 3d space are represented with a hat symbol. So unit 3 vectors will be represented as, say, m-hat.
If we prepared a system in n, and then moved to m-hat, the result is that <sigma-m> = n dot product m. This means that m inherits the results of n to some extent???? (Do more here).
and and or statements are different in quantum mechanics than in classical. Take the following scenario:
A: sigma-z = +1
B: sigma-x = +1
(A or B)
If someone else where to complete the A measurement and it was found to be +1, then it would continue to be +1 if we measured it. However if someone else were to complete the A measurement and it was found to be +1 but we instead chose to measure B it would erase/reset their A measurement and we might find A to be -1. (Is this right???). Therefore, the order of the experiment matters. In the first case the statement A or B is always true, but in the second ordering of events A or B has a 25% chance of being false.
General Readings and Resources
Seven Principles of Quantum Mechanics – https://arxiv.org/pdf/quant-ph/0212126.pdf
The ultimate reading list for beginners – https://dornsife.usc.edu/assets/sites/1045/docs/qmreading2018.pdf
Bra-ket Notation:
ket -> denotes a column vector |B> –
. Is the original vector space.
bra -> denotes a row vector that is the conjugate transpose (also called hermitian conjugate) of column vector??? <A| – (a1*,a2*). It is represented as a row vector simply to keep track of the fact that it belongs to the dual vector space rather than the single vector space of the ket vector.
bra-ket -> <A|B> means <A|⋅|B> which means the inner product of A and B.
<A|B> – <A| is (a1*,a2*) and |B> is then <A|B> represents the inner union which is the first entry of A multiplied by the first entry of B, and so on. a1*b1 + a2*b2.
<B|B> – <B| is (b1*,b2*) and |B> is then <B|B> represents the inner union which will be POSITIVE and a REAL NUMBER.
In application kets are associated with a specific state that can be measured. It can be up or down, Horizontal or Vertical, etc. Often these states are somewhat arbitrary in terminology because at such a small level Up and Down have little actual meaning, but there needs to be some terminology to differentiate them.
Two vectors <A|B> are orthogonal if the inner product is zero, <A|B> = 0.
Bra-ket Multiplication:
<B|M = page 59
Hermitiant Conjugation:
M|A> = |B>
Eigenstate/Eigenfunction: A wave function that, when acted upon by a general real-space operator, will multiply. In other words, it won’t ‘stretch’ in either direction. It scales uniformly.
Eigenvalue: The value by which the Eigenstate multiplies.
Self-Adjoint (Hermitian) Operator: ??? Represent physical observables in QM.
Born’s Rule:
Quantum Tomography: The measurement of the quantum state before measurement. As measuring will have a result on the system sometimes we are trying to figure out what that system looked like pre-measurement. This is what Quantum Tomography is for.
Schrodinger Equation:
The quantum mechanics version of Hamilton’s equations
The Problem of Time:
Ekatrine Moreva:
Time from quantum entanglement: an experimental illustration – https://arxiv.org/pdf/1310.4691.pdf
Quantum Time: experimental multi-time correlations – https://arxiv.org/pdf/1710.00707.pdf
Page and Wooters
Causation in QM:
Milburn and Shrapnel – https://arxiv.org/pdf/1809.03191.pdf
Wojciech Hubert Zurek:
Quantum Darwinism: https://arxiv.org/pdf/0903.5082.pdf
The idea is that there are pointer states that exist within a quantum system that can, or has a chance of, surviving when decoherence occurs.
Zurek also has intresting views on Maxwell’s demon and thermodynamic principles.
