Notepad/enter/Machine Tips (Quantum)/Resources/Concepts Review/Math/Math terms & Probabilities.md

• pi and the Tau Manifesto
• absolute probabilities : absolute probability of an event
• conditional probabilities : probability of one event given the knowledge that another event occurred
• joint probabilities : probability of two events occurring together

from math  import  asin, sqrt

def prob_to_angle(prob)

return 2*asin(sqrt(prob))

#specifiy marginal probability 
event_a = 0.4

#set qubit to prior 
qc.ry(prob_to_angle(event_a), 0)

run_circuit(qc)

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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Calculate the joint probability by multiplying both probabilities.

#specify the marginal probabilities  
event_a =  0.4
event_b = 0.8

qc = QuantumCircuit(4)

#set  qubit to prior 
qc.ry(prob_to_angle(event_a), 0)

#apply modifier 
qc.ry(prob_to_angle(event_b), 0)

run_circuit(qc)

state - combination of the qubits values

• let qubit represent marginal prob of an event

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More Uncertainty Concepts

tensor products : a way of calculating amplitudes

• Hamiltonian : operator corresponding to total energy of the system, both potential & kinetic, both energy spectrum & time-evolution
• spectrum : energy eigenvalues
• unit-norm vectors : probabilities normalized to 1
• pure states - physical pure states in QM represented as unit norm vectors in Hilbert space
• Time evolution - the application of an evolution operator in Hilbert vector space
• norm : behaves like the distance from the origin
• norm of a physical state should stay fixed in matrix mechanics so that the evolution operator is unitary, and that the operators can be represented as matrices
• operator : function over a space of physical states onto another space of physical space

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• differentiation : abstract function that accepts a function and returns another function
• differential operators : observables in wave mechanics representation of QM as wave function varies with space and time, or momentum & time
• observable : any quantity which can be measured in a physical experiment
  • associated with a self-adjoinng linear operator, x* = x,
  • must be Hermitian & yield real eigenvalues
• Hermitian : real symmetric matrices or a_{ij} = a_{ji}.
  • fundamental to quantum mechanics as these describe the operators with necessarily real eigenvalues

  •  \begin{bmatrix}
    

0 & {a - ib} & {c - id} \ {a + ib} & 1 & {m- in} \ {c + id} & {m + n} & 2 \end{bmatrix}

• discrete eigenstates : any state is a sum
• continuum of eigenstates : any state is an integral
• expectation value: average or mean value is average measurement of an observable for particle in region R.

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• Schwarzian derivative - non-linear differential operator similar to the invariant derivative under Mobius transformation.

g(z) = \frac{az + b}{cz + d} = 0

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.

• If you let f be a Schwartz function, then by inverse Fourier transform, P is exhibited as a Fourier multiplier.

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Fermi Quantities:

• Fermi temperature : $T_{F} = \frac{E_{F}}{k_{B}} where $k_B is Boltzmann constant, 1.38064 \cdot{10 ^{-23}}J \cdot{K^-1}
• Fermi momentum : $p_{F} = hk_{F}$ 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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• 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.

\psi(r) = e^{i k \cdot r} u(r) 

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. Functions of this form are known as Bloch functions (Bloch electrons or Bloch Waves) or Bloch states & underlie electronic band structure.

• 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?"

Bra-ket notation from wave machanics:

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• Del or Nabla : operator used in math for many purposes
  • for 1-D, standard derivative of the function defined in calculus
  • for fields or function defined on a multi-dimensional domain, may denote any one of 3 operators
  1. gradient: (locally) steepest slope of a scalar/vector field, or product with a scalar field or grad f = \nabla f
  2. divergence: the volume density of an outward flux of a vector field, dot product with the field
  3. curl (rotation) of a vector field, cross product with the field

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• Splines: special function defined by piecewise polynomials
  • Many applications in art, game development, computer science & computation!