- pi and the [Tau Manifesto](https://tauday.com/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. 


![[Pasted image 20230215042855.png]]

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 
---
#### 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](https://en.wikipedia.org/wiki/Operator_(physics)#Operators_in_quantum_mechanics) can be represented as matrices
- operator  : function over a space of physical  states onto another space of physical  space
---

- 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](https://en.wikipedia.org/wiki/Hermitian_matrix) 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. 

---

- **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. 

---

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. 

---

- 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. 
  
  [![](https://upload.wikimedia.org/wikipedia/commons/thumb/9/90/QuantumHarmonicOscillatorAnimation.gif/280px-QuantumHarmonicOscillatorAnimation.gif)](https://en.wikipedia.org/wiki/File:QuantumHarmonicOscillatorAnimation.gif)
- 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: 

![[Pasted image 20230517182829.png]]

---
- 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* 


---

- Splines: special function defined by piecewise polynomials 
	- Many applications  in art, [game development](https://www.youtube.com/watch?v=jvPPXbo87ds), computer science & computation![[Pasted image 20230802150926.png]]