123 lines
5.6 KiB
Markdown
123 lines
5.6 KiB
Markdown
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- pi and the [Tau Manifesto](https://tauday.com/tau-manifesto)
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- **absolute probabilities** : absolute probability of an event
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- **conditional probabilities** : probability of one event given the knowledge that another event occurred
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- **joint probabilities** : probability of two events occurring together
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```
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from math import asin, sqrt
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def prob_to_angle(prob)
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return 2*asin(sqrt(prob))
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#specifiy marginal probability
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event_a = 0.4
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#set qubit to prior
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qc.ry(prob_to_angle(event_a), 0)
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run_circuit(qc)
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```
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*note*: the ry gate similar to the H gate in that cuts the probability into two parts but provides us with tool to control the size of two parts.
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![[Pasted image 20230215042855.png]]
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Calculate the joint probability by multiplying both probabilities.
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```
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#specify the marginal probabilities
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event_a = 0.4
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event_b = 0.8
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qc = QuantumCircuit(4)
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#set qubit to prior
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qc.ry(prob_to_angle(event_a), 0)
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#apply modifier
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qc.ry(prob_to_angle(event_b), 0)
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run_circuit(qc)
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```
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state - combination of the qubits values
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- let qubit represent marginal prob of an event
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---
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#### More Uncertainty Concepts
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**tensor products** : a way of calculating amplitudes
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- **Hamiltonian** : operator corresponding to total energy of the system, both potential & kinetic, both energy spectrum & time-evolution
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- **spectrum** : energy eigenvalues
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- **unit-norm vectors** : probabilities normalized to 1
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- **pure states** - physical pure states in QM represented as unit norm vectors in Hilbert space
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- **Time evolution** - the application of an evolution operator in Hilbert vector space
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- **norm** : behaves like the distance from the origin
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- norm of a physical state should stay *fixed* in **matrix mechanics** so that the evolution operator is unitary, and that the [operators](https://en.wikipedia.org/wiki/Operator_(physics)#Operators_in_quantum_mechanics) can be represented as matrices
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- operator : function over a space of physical states onto another space of physical space
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---
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- differentiation : abstract function that accepts a function and returns another function
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- differential operators : observables in **wave mechanics** representation of QM as wave function varies with space and time, or momentum & time
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- observable : any quantity which can be measured in a physical experiment
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- associated with a self-adjoinng linear operator, x* = x,
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- must be Hermitian & yield real eigenvalues
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- Hermitian : real symmetric matrices or $a_{ij} = a_{ji}$.
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- [fundamental](https://en.wikipedia.org/wiki/Hermitian_matrix) to quantum mechanics as these describe the operators with necessarily real eigenvalues
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- $$ \begin{bmatrix}
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0 & {a - ib} & {c - id} \\
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{a + ib} & 1 & {m- in} \\
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{c + id} & {m + n} & 2
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\end{bmatrix}$$
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-
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- discrete eigenstates : any state is a sum
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- continuum of eigenstates : any state is an integral
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- expectation value: average or mean value is average measurement of an observable for particle in region R.
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---
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- **Schwarzian derivative** - non-linear differential operator similar to the invariant derivative under Mobius transformation.
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$$
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g(z) = \frac{az + b}{cz + d} = 0
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$$
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Mobius transformations are the only functions with this property. Schwarzian derivative precisely measures the degree to which a function fails to be a Mobius transformation.
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- If you let f be a Schwartz function, then by inverse Fourier transform, P is exhibited as a Fourier multiplier.
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---
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Fermi Quantities:
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- Fermi temperature : $$T_{F} = \frac{E_{F}}{k_{B}}$$ where $k_B$ is Boltzmann constant, $1.38064 \cdot{10 ^{-23}}J \cdot{K^-1}$
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- Fermi momentum : $$p_{F} = hk_{F}$$
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where $k_{F}$ is the radius of fermi sphere and is called fermi wave vector. These quantities are not well-defined in cases where Fermi surface is non-spherical.
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---
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- Bloch's Theorem discovered in 1929 and state solutions to the Schrodinger equation in a periodic potential taking the form of a plane wave modulated by a periodic function.
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$$\psi(r) = e^{i k \cdot r} u(r) $$
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where $r$ is position, $\psi$ is the wave function, $u$ is a periodic function with the same periodicity as the crystal, the wave vector $k$ is the crystal momentumvector, $e$ is Eurler's number, and $i$ is the imaginary unity.
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Functions of this form are known as Bloch functions (Bloch electrons or Bloch Waves) or Bloch states & underlie electronic band structure.
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[](https://en.wikipedia.org/wiki/File:QuantumHarmonicOscillatorAnimation.gif)
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- Bloch's Theorem goes into spectral geometry which is a field in mathematics concerning relationships between geometric structures of manifolds & spectra of canonically defined differential operators. "Can one hear the shape of a drum?"
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-
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Bra-ket notation from wave machanics:
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![[Pasted image 20230517182829.png]]
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---
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- Del or Nabla : operator used in math for many purposes
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- for 1-D, standard derivative of the function defined in calculus
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- for fields or function defined on a multi-dimensional domain, may denote any one of 3 operators
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1. gradient: (locally) steepest slope of a scalar/vector field, or *product with a scalar* *field* or $grad f = \nabla f$
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2. divergence: the volume density of an outward flux of a vector field, *dot product with the field*
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3. curl (rotation) of a vector field, *cross product with the field*
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---
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- Splines: special function defined by piecewise polynomials
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- Many applications in art, [game development](https://www.youtube.com/watch?v=jvPPXbo87ds), computer science & computation![[Pasted image 20230802150926.png]]
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