MIT Quantum mechanics

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MIT Quantum mechanics lectures

 https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2016/resources/quantum-mechanics-as-a-framework/


Lecture 1: An overview of quantum mechanics.


1) Lineality         EOM       dynamical variables



linear equation  L u = 0  ( where L = linear operation,  u = unknown)

lineality 



examples of linearity, how you can separate L for Lu = 0

L is linear in this case.


Motion in 1D

m d^2x(t)/ dt^2 = - V ' (x(t))

* V(x)   particle moving in 1D x with under the influence of a potential.

* X(t)

V' derivative of V w,r,t to argument

this is newton's equation and is difficult to solve cause it's not an linear equation.

Derivative of a potential


if potential goes left then it's minus.

V' is the derivative of V with respect to argument.


* But Quantum mechanics is linear!

Schrodinger equation.


i h bar partial derivative with respect to time of psi is eqaul to H hat of psi, where H hat is called Hamiltonian and it's a linear operator.


This is Schrodinger equation.

L psi = 0 

L psi would be defined as i h bar d psi/ d t - H ^ psi


if particle is moving on the line, 


2) complex numbers




normal of z is a real number.

Example, what is the number z in unit radius.


 * z = cosθ + i * sinθ


* in quantum mechanics, psi is a complex number.

* Maxwell has defined, the norm of psi as the probabilities



Photons and the loss of determinism

3) Determinism

it all begins with photons, 

light / photons :     wave / particles

Einstein realized, for a photon, energy E is given by h nu.

where nu is the frequency of the light.


frequency times wave length is the velocity of light


Polarizer. 

if send light, linearly polarized along the x-axis, it all goes through,

and if send light linearly polarized along the y-axis, it goes nothing. 



The electric field of the alpha is E0 * cos α * X^ + E0 + sin α * Y^ . 


After polarizer E = E0 cos α X^

* The energy on an electromagnetic field is proportional to the magnitude of the electric field square.

* fraction of energy through (cos α)*(cos α)

Einstein realized that the photon is not identical it has a hidden value that we don't know about.

That's why we stuck with probabilities.

And it's hidden variable theory.

* John bell figured out that Quantum mechanics cannot be made deterministic with hidden variables.

So we lost determinism we could only get probability.

* Photon : either get through or not.

            Can only predict probabilities.

Dirac invented a notation like

        | photon; x>        this is photon polarized on x.

        I photon; y>         this is photon polarized on y.

** | photon; α> =   cos α | photon; x> + sin α | photon; y>

-> after the polarizer only |photon; x>


The nature of superposition. Mach-Zehnder interferometer.

4) Nature of superposition

Mach-Zehnder interferometer (1891 - 1892)

set of beam splitters and mirrors to determine the interference of light. 

Interferometer

* When we get light interference, each photon is interfering with itself. 

    They cannot interfere with each other.

    And therefore each photon will have to be in both beams at the same time.


Superposition

Single photon state = superposition of the photon in their upper beam and the photon in their lower beam.

Original vector

Superposition of |A> , and |B>

measure same property on |A> get "a"

measure same property on |B> get "b" 

* suppose have a quantum mechanical state, and state is ( α, β ∈ complex number )

    α|A> + β|B> in quantum mechanics " never get in between of a and b"

                "will either get little a or b with different probabilities "

                roughly speaking the probability(a) ~ |α|²    if get  "a" state becomes |A>

                                                  probability(b) ~ |β|²   if get "b" state becomes |B>


_____________________________________________________________________________________________

Lecture 2: Overview. Interaction-free measurements

1) More on superposition, the General state of a photon, and spin states.


If you have a state and superpose it then you haven't done anything.

The superposition of a state with itself has no physical import.







    








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