14. Quantum Mechanics

QM for Everyone is the experimental foundation.

In QM, Sage has no specific quantum module, so use SymPy’s quantum module: operators, commutators, wavefunctions, quantum states (eg, harmonic oscillator, spin systems). And Wolfram Quantum Framework

Categories of Physics	low speed	high-speed/energy
macro size/scale	Classical Mechanics - motion of cars, planets - engineering like buildings, transportation	Relativistic Mechanics - things near black holes or GPS satellites - Cosmology
micro size/scale	Quantum Mechanics - atoms, electrons - semiconductors, quantum computing	Quantum Field Theory - atom nucleus or particle collisions in accelerators - Particle Physics, Condensed matter(superconductors)

• Parallels between CM and QM

C Mechanics	QM
Statistics Mechanics	quantum field theory

14.1. QM for Everyone

14.1.1. Advanced Quantum Mechanics with Spins

Notes

• Two-Slit Experiment with Stern-Gerlach Analyzer Loop : There are two results using classical and quantum probability. The quantum probability result is correct where since the path taken is indeterminate(you can also say it takes both paths), the atom’s state does not change.

  • Detector Variant : If a detector is put in the analyzer loop, the atom’s x state becomes known meaning it no longer has a state in the z direction.

14.1.2. Intro & Quantum Probability

• Fermions : Metals conduct electricity do so because there is a “sea of electrons”. This is created from the Pauli exclusion principle, since electrons are fermions, and fermions stay apart, so they stay apart.

• Alpha Particles : Created by radioactive elements. Since it is slow and heavy, it has low penetration power but can damage biological tissue. Smoke detectors use heavy elements to detect smoke as when there is smoke the current flow between the 2 metal plates that are held at different voltages the current is changed.

• Magnetic Fields : These are vector fields that are created by magnets.

  • The closer to a magnet you are, the stronger the intensity.

  • Normally the arrows indicate where the north pole of a magnet will be attracted to.

  • Forces in the magnetic field can cause the magnet to move because of a net force or spin because of forces in opposite directions at different ends.

    • The net force can be calculated by finding the axis of increasing intensity and then finding the projection of the magnet onto the axis. When doing this it is useful to keep the south pole of the magnet on the axis.

  • Fat Arrows : For some magnetic fields, they can be represented by fat arrows where the width is the intensity of the magnetic field and the direction of the arrows is indicated by the fat arrow’s direction.

  • Magnetic Needles : These are idealized small magnets like those in compasses.

• Precession : A rotation that occurs when there is a magnet and a magnetic field.

  • Even when there is precession, the projection remains constant as it rotates perpendicular to the axis of increasing field.

• Stern-Gerlach Experiment : Atoms are injected through a region of space where because of a magnetic field, there will be precession. 2 magnets will deflect the atom until it hits the screen where there is a detector.

  • In the quantum world, since everything is random, the atoms’ chance of exiting a specific exit for positive and negative can be calculated using a formula.

• Birthday Problem : The problem asks you that how many people must be in a large room for the chance of at least 2 people having the same birthday to be a specific number.

  • There are many variants.

# Birthday Problem
result=False
count=0
for i in range(100000):
    result_in=False
    people=23
    birthdays=[13,101,155,165]
    for i in range(people-4):
        birthdays.append (randint(1,365))
    birthdays.sort()
    for i in range(1,366):
        if birthdays.count(i)>2:
            result=True
            result_in=True
    if result_in==True:
        count+=1
count

1181

# Penney's Game
me_win=0
comp_win=0
me_coins=[0,1,0]
comp_coins=[0,1,1]
for i in range(100000):
    coins=[2,2,2]
    coins_req=[coins[-3],coins[-2],coins[-1]]
    while coins_req!=me_coins and coins_req!=comp_coins:
        coins.append(randint(0,1))
        coins_req=[coins[-3],coins[-2],coins[-1]]
    if me_coins==coins_req:
        me_win+=1
    else:
        comp_win+=1
s(me_win,comp_win)

\[\displaystyle 50142 \qquad 49858 \qquad \]

# ???
s(11/12*10/12*9/12*8/12*7/12,1728-385)
plot(cos(x/2)^2,(x,0,2*pi), figsize=3)

\[\displaystyle \frac{385}{1728} \qquad 1343 \qquad \]

x=1
n=0
m=364
while x>0.5:
    x*=m
    m-=1
    x/=365
    n+=1
s(n)

\[\displaystyle 22 \qquad \]