Quantum Physics - High School

Notes taken during a 4-day long trip with my high school teacher to Perzanowo (2022)

Based on the following notes - Perzanowo 2022.pdf (in Polish)

1935 - Einstein, Podolsky, Rosen find that the quantum mechanics is incomplete, and requires refinement - EPR Paradox

Two-Electron System

Every electron contains a property called a spin - intrinsic angular momentum.

Pauli exclusion principle

No two electrons can occupy the same state

\require e n c l o s e {\overset{A}{↑} \overset{B}{↓}} + {\overset{A}{↓} \overset{B}{↑}}

Two close electrons have opposite spins

Let's send one electron from a pair to a person A, and the second one to the person B. During the travel, both electrons are in superposition (we don't know the exact, current spin). If person A checks the spin, then person B is 100% sure that their spin will be exactly the opposite
In theory, we can send information faster than speed of light this way

Hidden variables - variable (i.e. spin), which is unknown (i.e. spin during travel). Knowing the variable would let us determine the spin state, without measuring it

1964, John Bell - There exists a measurable quantity which, if hidden variables exist, satisfies an inequality $$C \leq 2$$If there are no hidden values, $C$ can exceed than 2

The Quantum World

We analize state of the system, and explore its evolution
Let's consider a model system of a single isolated spin
The spin space is a two-dimensional space in which there are two basis states
${| + ⟩, | - ⟩} \leftarrow$ 2 basis states, spin-up and spin-down vectors
$| v ⟩$ is a ket-vector notation

Any state in this space, is a superposition of the basis states

| Ψ ⟩ = Ψ_{+} | + ⟩ + Ψ_{-} | - ⟩

When measuring the spin, we can only obtain one of the two values:

+ \frac{1}{2} ℏ or - \frac{1}{2} ℏ, where ℏ = \frac{h}{2 π}

Measured values (here - spin) are represented by operator. Operator connected with the spin measurement along the z-axis is denoted by $σ_{z}$

Operators affect the vectors. If the system is in $| +_{z} ⟩$ state, then using the operator $σ_{z} | + z ⟩ = \frac{1}{2} ℏ | + z ⟩$ - $\frac{1}{2} ℏ$ is the measured value
Similarly - $σ_{z} | -_{z} ⟩ = \frac{1}{2} ℏ | -_{z} ⟩$ - $- \frac{1}{2} ℏ$ is the measured value

Coming back to the spin-state vector in the basis notation ${| + ⟩, | - ⟩}$

| Ψ ⟩ = Ψ_{+} | +_{z} ⟩ + Ψ_{-} | -_{z} ⟩

$Ψ_{+}$ and $Ψ_{-}$ coefficients are related to the probability of getting measurement values equal to $+ \frac{1}{2} ℏ$ and $- \frac{1}{2} ℏ$ respectively
The probabilities:

P_{+} = | Ψ_{+} |^{2} and P_{-} = | Ψ_{-} |^{2}

Because those are the only measurements, then $P_{+} + P_{-} = 1$

Examples

$| Ψ ⟩ = 2 | +_{z} ⟩ - 3 | -_{z} ⟩$ is impossible, as $P = 2^{2} + (- 3)^{2} = 4 + 9 = 13 > 1$
$| Ψ ⟩ = \frac{1}{2} | +_{z} ⟩ + \frac{\sqrt{3}}{2} | -_{z} ⟩$ is good, as $P = {(\frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2} = \frac{1}{4} + \frac{3}{4} = 1$

Continuing

Let's rotate the measurement tool and measure the spin on the x-axis
For simplicity, let's take the unit of spin to be $\frac{1}{2} ℏ$
We measure the spin, and we get value 1. A repeated measurement of that spin will yield the same result
Measurement reduces the state to the one corresponding to the measured value

Measuring the spin along x-axis, we get value $- 1$ . The spin gets settled during the measurement
New basis vectors can be measured, and knowing that the spin plane is two-dimensional, $| +_{x} ⟩$ is expressed by $| +_{z} ⟩$ and $| -_{z} ⟩$ vectors:

| +_{x} ⟩ = \frac{1}{\sqrt{2}} | +_{z} ⟩ + \frac{1}{\sqrt{2}} | -_{z} ⟩

From this, we see $P_{+} = | Ψ_{+} |^{2} = {(\frac{1}{\sqrt{2}})}^{2} = \frac{1}{2}$ - 50/50 chance between + and -
For $| -_{x} ⟩$ :

| -_{x} ⟩ = \frac{1}{\sqrt{2}} | +_{z} ⟩ - \frac{1}{\sqrt{2}} | -_{z} ⟩

These vectors are orthogonal to the spin space
Vectors $| +_{x} ⟩$ and $| -_{x} ⟩$ are represented by

\begin{aligned} | +_{x} ⟩ = [\begin{array}{c} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{array}] & | -_{x} ⟩ = [\begin{array}{c} \frac{1}{\sqrt{2}} \\ - \frac{1}{\sqrt{2}} \end{array}] \end{aligned}

And calculating the scalar product: $\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} \cdot \frac{1}{- \sqrt{2}} = 0$

Let's rotate the measuring device to measure the spin along the y-axis. We get value 1, being the spin projected onto the state $| +_{y} ⟩$
But for value -1, new vectors appear, but with the spin space being two-dimensional, vectors $| +_{y} ⟩$ and $| -_{y} ⟩$ must be denoted by $| +_{z} ⟩$ and $| -_{z} ⟩$

Ψ | +_{y} ⟩ = Ψ_{+} | +_{z} ⟩ + Ψ_{-} | -_{z} ⟩

with such coefficients that $| Ψ_{+} |^{2} = | Ψ_{-} |^{2} = \frac{1}{2}$

But we already used the basic set of values:

\begin{array}{r} Ψ_{+} = \frac{1}{\sqrt{2}}, Ψ_{-} = \frac{1}{\sqrt{2}} \to | +_{x} ⟩ \\ Ψ_{+} = \frac{1}{\sqrt{2}}, Ψ_{-} = - \frac{1}{\sqrt{2}} \to | -_{x} ⟩ \end{array}

Let's introduce a number, which square root equals $- 1$ , and denote it as $i$
Now:

\begin{array}{r} | +_{y} ⟩ = \frac{1}{\sqrt{2}} | +_{z} ⟩ + \frac{i}{\sqrt{2}} | -_{z} ⟩ \\ | -_{y} ⟩ = \frac{1}{\sqrt{2}} | +_{z} ⟩ - \frac{i}{\sqrt{2}} | -_{z} ⟩ \end{array}

The number $i$ belongs to set of complex numbers

Complex Numbers

A complex number is pair of real numbers, equipped with operations $(+, \cdot)$ , identity elements of those operations, and distributive and associative laws.
Addition

(x_{1}, y_{1}) + (x_{2}, y_{2}) = (x_{1} + x_{2}, y_{1} + y_{2})

Multiplication
Let's create a complex plain, on which a point represents a complex number
The representation of the point: $(x, y) \to 1 \cdot x + i \cdot y = x + i y$

\begin{aligned} \begin{array}{ll} Z_{1} = x_{1} + i y_{1} \\ Z_{2} = x_{2} + i y_{2} \end{array}} Z_{1} \cdot Z_{2} & = (x_{1} + i y_{1}) (x_{2} + i y_{2}) = x_{1} x_{2} + i x_{1} y_{2} = i x_{2} y_{1} - y_{1} y_{2} \\ = (x_{1} x_{2} - y_{1} y_{2}) + i (x_{1} y_{2} + x_{2} y_{1}) = x + i y = Z \end{aligned}

Remembering that $i^{2} = - 1$

Let's take a look to $Z = i = 0 + i \cdot 1$
We can see - $i^{2} = - 1$ , $i^{3} = - i$ , $i^{4} = 1$

Multiplying $i$ to consecutive powers rotates the vector representive the complex number. The angle between the vector and Realis axis is increasing

\begin{aligned} i : \frac{π}{2} \\ i^{2} : π = \frac{π}{2} + \frac{π}{2} \\ i^{2} : \frac{3}{2} π = π + \frac{π}{2} \\ i^{2} : 2 π = \frac{3}{2} π + \frac{π}{2} \end{aligned}

Multiplication can be represented as a sum of angles. Let's recognize, that this closely resembles the something we already know:

\begin{aligned} i = e^{i \frac{π}{2}} \\ i = e^{i \frac{π}{2} \cdot 2} \\ i = e^{i \frac{π}{2} \cdot 3} \end{aligned}

The multiplicaiton $\frac{π}{2} \cdot n$ tells us the angle between the vector representing the complex number and the realis axis

Any complex number can be expressed as $Z = | Z | e^{i ψ}$ , where $| Z |$ is the modulus of the complex number - length of the vector representing the number

\begin{aligned} z = x + i y \to | z | = \sqrt{x^{2} + y^{2}} \\ | z |^{2} = x^{2} + y^{2} = (x + i y) (x - i y) \end{aligned}

Numbers $z_{1} = x + i y$ and $z_{2} = x - i y$ are complex conjugates of each other, denoted as $z^{*}$
We can also find that: $(z^{*})^{*} = ((x + i y)^{*})^{*} = (x - i y)^{*} = (x + i y) = z$

Three ways of complex number notation:

Algebraic - $z = x + i y$
Exponential - $z = | z | e^{i ψ}$ , $| z | = \sqrt{x^{2} + y^{2}}$ , $x = | z | \cos ψ$
Trigonometric - $z = | z | (\cos ψ + i \sin ψ) = | z | \sin ψ$ , $y = | z | \sin ψ$

Euler's formula - $e^{i ψ} = \cos ψ + i \sin ψ$
And the exciting stuff! - $e^{i π} + 1 = 0$

Excercises

By doing the calculations, we have shown the following:

\begin{array}{r} σ_{z} | +_{x} ⟩ = | -_{z} ⟩ \\ σ_{z} | -_{x} ⟩ = | +_{z} ⟩ \end{array}

Let's check what will be the result of $σ_{x} | +_{z} ⟩$
Expressing vectors $| +_{z} ⟩$ and $| -_{z} ⟩$ using vectors $| +_{x} ⟩$ and $| -_{x} ⟩$ :

\begin{array}{r} {\begin{cases} | +_{x} ⟩ = \frac{1}{\sqrt{2}} | +_{z} ⟩ + \frac{1}{\sqrt{2}} | -_{z} ⟩ \\ | -_{x} ⟩ = \frac{1}{\sqrt{2}} | +_{z} ⟩ - \frac{1}{\sqrt{2}} | -_{z} ⟩ \end{cases} \\ (+) | +_{x} ⟩ + | -_{x} ⟩ = \sqrt{2} | +_{z} ⟩ \\ (-) | +_{x} ⟩ - | -_{x} ⟩ = \sqrt{2} | -_{z} ⟩ \end{array}

And so

{\begin{cases} | +_{z} ⟩ = \frac{1}{\sqrt{2}} | +_{x} ⟩ + \frac{1}{\sqrt{2}} | -_{x} ⟩ \\ | -_{z} ⟩ = \frac{1}{\sqrt{2}} | +_{x} ⟩ - \frac{1}{\sqrt{2}} | -_{x} ⟩ \end{cases}

Doing the calculations

\begin{aligned} σ_{x} | +_{z} ⟩ & = \frac{1}{\sqrt{2}} σ_{x} | +_{x} ⟩ + \frac{1}{\sqrt{2}} | -_{x} ⟩ = \frac{1}{\sqrt{2}} | +_{x} ⟩ - \frac{1}{\sqrt{2}} | -_{x} ⟩ = | -_{z} ⟩ \\ σ_{x} | -_{z} ⟩ & = | +_{z} ⟩ \end{aligned}