NON-DUALITY
Notes:
1) FUZZINESS: The Standard model which explains all this (currently our best quantum theory of physics), explains these are point particles with NO VOLUME taking up NO SPACE. What appears as volume is the fuzzy probability patterns of the nucleus and especially electrons. But really nothing is there until we observe it.
2) NO OBJECTS: This further helps to see the illusion of matter and the veil of separation. At the quantum level there are NO OBJECTS. Nothing that exists separate and outside of us until measured. And even when we do observe them into creation, they are all identical at the most fundamental level!
3) ONENESS: If we go all the way back to the Big Bang itself, all the forces merge into a unified superforce and all particles merge in a quark-electron plasma. Which then ultimately in superstring theory, forces and particles even merge into a simplistic Oneness. Scientific materialism actually preaches the Oneness of the Material universe at its roots, but has no satisfactory explanation to explain how complex life emerges.
4) FRAMEWORK: Using the materialistic framework of quantum theory, How could something so radically simple become something to unfathomably complex (our Universe and Life in it)?. How can something unreal create Reality as we know it (especially by chance)? It makes no sense in a material framework. Quantum field theory and Darwinian evolution is not enough.
1) FUZZINESS: The Standard model which explains all this (currently our best quantum theory of physics), explains these are point particles with NO VOLUME taking up NO SPACE. What appears as volume is the fuzzy probability patterns of the nucleus and especially electrons. But really nothing is there until we observe it.
2) NO OBJECTS: This further helps to see the illusion of matter and the veil of separation. At the quantum level there are NO OBJECTS. Nothing that exists separate and outside of us until measured. And even when we do observe them into creation, they are all identical at the most fundamental level!
3) ONENESS: If we go all the way back to the Big Bang itself, all the forces merge into a unified superforce and all particles merge in a quark-electron plasma. Which then ultimately in superstring theory, forces and particles even merge into a simplistic Oneness. Scientific materialism actually preaches the Oneness of the Material universe at its roots, but has no satisfactory explanation to explain how complex life emerges.
4) FRAMEWORK: Using the materialistic framework of quantum theory, How could something so radically simple become something to unfathomably complex (our Universe and Life in it)?. How can something unreal create Reality as we know it (especially by chance)? It makes no sense in a material framework. Quantum field theory and Darwinian evolution is not enough.
2 State Quantum System - like spin up and spin down (or left slit / right slit), or simple hydrogen atom
Imagine Observable A associated with some physical property and this is its eigenvalue equation.
with 2 eigenstates. λn is the eigenvalues and 'u'n are the eigenstates (both orthogonal and normalized)
Eigenstates are the possible measurement outcomes for a given Observable, usually +1 and -1 for a two state system (or 1 & 0), Both
Build a quantum state |Ψ> that is a linear superposition between these two states or by its Ket |S>
|c1|^2 + |c2|^2 = 1 ... normalized
Imagine Observable A associated with some physical property and this is its eigenvalue equation.
with 2 eigenstates. λn is the eigenvalues and 'u'n are the eigenstates (both orthogonal and normalized)
Eigenstates are the possible measurement outcomes for a given Observable, usually +1 and -1 for a two state system (or 1 & 0), Both
Build a quantum state |Ψ> that is a linear superposition between these two states or by its Ket |S>
|c1|^2 + |c2|^2 = 1 ... normalized
Now consider what happens when we measure property A in state |Ψ>
The only possible outcome of a measurement A is one of its eigenvalues
Postulate of State Collapse of QM: Right after the measurement, the state of the system changes from |Ψ> --> u1 or u2 with eigenvalue λ1 or λ2 (see below). These are the two possible measurement outcomes.
c1 and c2 are complex amplitudes for each possible state
These numbers come from the results of measurement outcomes (Q.M. is a calculus for determining measurement outcomes.
The only possible outcome of a measurement A is one of its eigenvalues
Postulate of State Collapse of QM: Right after the measurement, the state of the system changes from |Ψ> --> u1 or u2 with eigenvalue λ1 or λ2 (see below). These are the two possible measurement outcomes.
c1 and c2 are complex amplitudes for each possible state
These numbers come from the results of measurement outcomes (Q.M. is a calculus for determining measurement outcomes.
See below 2 Frameworks.... to compare pure and mixed states.
1st framework is Superposition State (Above) - One quantum object
2nd framework (can happen in Classical) is a Statistical Mixture of States... Ensemble reflecting same probability of Superposition
System is now made of many copies of |u1> and many copies of |u2>
n1 copies of |u1> and n2 copies of |u2>
second step is selecting a state at random from this collection of states or mixture.
third step is the measurement... because |u1> and |u2> are eigenstates of the system we are measuring,
we know with certainty what the result of the measurement will be.
1st framework is Superposition State (Above) - One quantum object
2nd framework (can happen in Classical) is a Statistical Mixture of States... Ensemble reflecting same probability of Superposition
System is now made of many copies of |u1> and many copies of |u2>
n1 copies of |u1> and n2 copies of |u2>
second step is selecting a state at random from this collection of states or mixture.
third step is the measurement... because |u1> and |u2> are eigenstates of the system we are measuring,
we know with certainty what the result of the measurement will be.
THE INTERFERENCE TERM IS KEY TO QUANTUM MECHANICS!! (below)
Double Slit experiment can only be understood with Superposition terms.
Double Slit experiment can only be understood with Superposition terms.
Sometimes we don't have enough information of our system at the state selection step... perhaps a mixture of up and down photons sent through polarization but we don't know initial condition.
Then we need both probabilistic steps.
If we know everything there is to know about our quantum system, the probability is a purely quantum effect that arises when we perform measurements. When this happens we say that we have pure states.
If we only have partial information about our system then probability enters in 2 places.
Good example is when we have a macroscopic system on the order of avogadros number of particles.
We start with the probability of statistical mechanics.
For example a system at thermodynamic equilibrium. Probability of particle being in energy state E at temperature T is given by below.
Probabilities below tells us how likely each pure state is... (sum of all p's = 1).
The p probabilities are independent of quantum mechanics and simply encode our lack of knowledge of the state of a system.
Typically quantum mechanics taught only with pure states.
Mixed states important for studying macroscopic systems with many particles.
Then we need both probabilistic steps.
If we know everything there is to know about our quantum system, the probability is a purely quantum effect that arises when we perform measurements. When this happens we say that we have pure states.
If we only have partial information about our system then probability enters in 2 places.
Good example is when we have a macroscopic system on the order of avogadros number of particles.
We start with the probability of statistical mechanics.
For example a system at thermodynamic equilibrium. Probability of particle being in energy state E at temperature T is given by below.
Probabilities below tells us how likely each pure state is... (sum of all p's = 1).
The p probabilities are independent of quantum mechanics and simply encode our lack of knowledge of the state of a system.
Typically quantum mechanics taught only with pure states.
Mixed states important for studying macroscopic systems with many particles.
Density Operator in context pure states leads to an equivalent formalism to that of state vectors.