Reading Week 2 - Intro to Bracket Notation

2.2: Complex Numbers

There's a lot of complex number math that I'm familiar with, but here's an overview.

$\sqrt{- 1} = i$
$a + b i = z$
$a = r \cos (θ)$
$b = r \sin (θ)$
$z = r (\cos (θ) + i \sin (θ)) = r e^{i θ}$
(De Moivre's Theorem) $z^{n} = (r (\cos (θ) + i \sin (θ)))^{n} = r^{n} (\cos (n θ) + i \sin (n θ))$
(Complex Conjugate) $z^{*} = a - b i$
(Norm) $| z | = \sqrt{a^{2} + b^{2}}$
$| z | = z z^{*}$
$| z |^{2} = r^{2} = (r e^{i θ}) (r e^{- i θ})$

2.5: Linear Vector Spaces

Here, we learn all about the $| ⟩$ bracket notation. Given by \bracket{} in latex:

| 0 ⟩

A lot of this is review for me, see Chapter 1 - Vector Spaces for more info on what's covered here. In short, we have the following:

A linear vector space $V$ is a collection of vector that meet the following:
- $| X ⟩ + | Y ⟩ = | Y ⟩ + | X ⟩$ (commutativity and closure)
- (associative property)
- There's a null or zero vector where $| 0 ⟩ + | X ⟩ = | X ⟩$ for any $| X ⟩ \in V$
- Each vector $| X ⟩$ has an additive inverse $| - X ⟩$ such that $| X ⟩ + | - X ⟩ = | 0 ⟩$
There's scalar multiplication with distributive properties:
- $a (| X ⟩ + | Y ⟩) = a | X ⟩ + a | Y ⟩$
- $a (b | X ⟩) = a b | X ⟩$
- There's a multiplicative identity 1 where $1 | X ⟩ = | X ⟩$
- $0 | X ⟩ = | 0 ⟩$

The scalars belong to some field $F$ which essentially just denotes the reals $R$ or the complex numbers $C$ .

A set of these vectors is linearly independent if the following sum only is satisfied by all $a_{i} = 0$ :

\sum_{i = 1}^{n} a_{i} | v_{i} ⟩ = | 0 ⟩

Otherwise the vectors are linearly dependent and if we know say $a_{1} \neq 0$ then we know that:

| v_{1} ⟩ = \frac{- a_{2} | v_{2} ⟩ - \dots - a_{n} | v_{n} ⟩}{a_{1}}

A vector space of $n$ linearly-independent vectors is said to have dimension $n$ . It's written $V^{n}$ . A vector $| V ⟩$ in $n$ -dimensional vector space can be written as a linear combination of $n$ linearly independent vectors ${| v_{1} ⟩, . . ., | v_{n} ⟩}$ . This set of $n$ linearly independent vectors is called a basis:

| V ⟩ = \sum_{i = 1}^{n} a_{i} | v_{i} ⟩

Each $a_{i}$ is the component of the state vector $| V ⟩$ on the basis $| v_{i} ⟩$ . For example, the standard basis for $R^{3}$ are the vectors:

\hat{i} = (1, 0, 0), \hat{j} = (0, 1, 0), \hat{k} = (0, 0, 1)

We can add two of these vectors where:

| V_{1} ⟩ = \sum_{i = 1}^{n} a_{i} | v_{i} ⟩, | V_{2} ⟩ = \sum_{i = 1}^{n} b_{i} | v_{i} ⟩ \Rightarrow | V_{1} ⟩ + | V_{2} ⟩ = \sum_{i = 1}^{n} (a_{i} + b_{i}) | v_{i} ⟩

and scaled by some $α \in F$ :

α | V ⟩ = \sum_{i = 1}^{n} (α a_{i}) | v_{i} ⟩

Inner Product

The inner product takes in two vectors $| X ⟩, | Y ⟩$ and outputs a scalar, denoted $⟨ X | Y ⟩$ . Here:

$⟨ X | Y ⟩ = {⟨ Y | X ⟩}^{*}$
$⟨ X | X ⟩ \geq 0$ and is only equal iff $| X ⟩ = | 0 ⟩$
$⟨ α X | Y ⟩ = α ⟨ X | Y ⟩$
$⟨ X + Y | Z ⟩ = ⟨ X | Z ⟩ + ⟨ Y | Z ⟩$ (additivity in the first slot)
Additivity in the second slot
Additivity in both slots

Two vectors are orthogonal if $⟨ X | Y ⟩ = 0$ .

The norm of a vector $| V |$ is given by:

\sqrt{⟨ V | V ⟩} = | V |

If the norm is 1, then $| V ⟩$ is normalized.

If a set of basis vectors are normalized and are orthogonal to one another, they are an orthonormal basis.

Assume $| V_{a} ⟩ = \sum_{i = 1}^{n} a_{i} | v_{i} ⟩$ and similarly $| V_{b} ⟩ = \sum_{i = 1}^{n} b_{i} | v_{i} ⟩$ . The inner product of the two vectors is:

⟨ V_{a} | V_{b} ⟩ = \sum_{i} \sum_{j} a_{i}^{*} b_{j} ⟨ v_{i} | v_{j} ⟩

Where since all the $v_{i}, v_{j}$ 's are orthonormal to each other, then only:

⟨ v_{i} | v_{j} ⟩ = {\begin{cases} 1 & i = j \\ 0 & i \neq j \end{cases} = δ_{i j}

This function $δ_{i j}$ is the Kronecker delta function. Applying this, we get that:

⟨ V_{a} | V_{b} ⟩ = \sum_{i = 1}^{n} a_{i}^{*} b_{i}

We know that $| V_{a} ⟩, | V_{b} ⟩$ are given by their components:

| V_{a} ⟩ = (a_{1}, . . ., a_{n}), | V_{b} ⟩ = (b_{1}, . . ., b_{n})

So then:

⟨ V_{a} | V_{b} ⟩ = [\begin{matrix} a_{1}^{*} & \dots & a_{n}^{*} \end{matrix}] [\begin{matrix} b_{1} \\ ⋮ \\ b_{n} \end{matrix}]

If we instead have some continuous functions $a_{i} = ψ_{a} (x), b_{i} = ψ_{b} (x)$ then:

⟨ V_{a} | V_{b} ⟩ = \int ψ_{a}^{*} (x) ψ_{b} (x) d x

2.6: Using Dirac's Bra-ket Notation

We've seen writing an abstract vector $| V ⟩$ as a column vector in basis $| v_{i} ⟩$ . We write $| V ⟩$ as:

| V ⟩ = [\begin{matrix} a_{1} \\ \dots \\ a_{n} \end{matrix}]

The conjugate transpose vector is $⟨ V |$ on the basis $⟨ v_{i} |$ writing it as:

⟨ V | = [\begin{matrix} a_{1}^{*} & \dots & a_{n}^{*} \end{matrix}]

So for every ket vector $| V ⟩$ there's a corresponding bra vector $⟨ V |$ and vice versa. Together, these vectors form dual vector spaces.

Here we are assuming that:

| v_{i} ⟩ = [\begin{matrix} 0 \\ ⋮ \\ 0 \\ 1 \\ 0 \\ ⋮ \\ 0 \end{matrix}]

at the corresponding $i$ -th position. As such, then:

| V ⟩ = a_{1} [\begin{matrix} 1 \\ 0 \\ ⋮ \\ 0 \end{matrix}] + a_{2} [\begin{matrix} + 0 \\ 1 \\ ⋮ \\ 0 \end{matrix}] + \dots + a_{n} [\begin{matrix} 0 \\ 0 \\ ⋮ \\ 1 \end{matrix}]

Missed Lecture Slides: Hamiltonian

The evolution of a wave function with time is governed by the time-dependent Schrodinger equation:

i ℏ \frac{\partial}{\partial t} Ψ (x, t) = H Ψ (x, t)

Here $H$ is the Hamiltonian and is the energy operator. Namely:

H = \underset{Kinetic energy Operator}{\underset{⏟}{- \frac{ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}}}} + \underset{potential energy operator}{\underset{⏟}{U (x)}}

In contrast, we have the time-independent Schrodinger equation in 3D:

\nabla^{2} ψ (\vec{r}) = \frac{2 m}{ℏ^{2}} (U (\vec{r}) - E) ψ (\vec{r})

with $\vec{r} = (x, y, z)$ position and $\nabla$ is the Laplace Operator:

\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}

In spherical coordinates it becomes:

![[Physics CPE 345 Quantum Computing Lecture slides Quantum superposition 240408.pdf#page=17]]

The general solution for the wave function in hydrogen is:

Ψ_{n l m} (\vec{r}, t) = R_{n l} (r) Y_{l m} (θ, ϕ) e^{- i \frac{E_{n}}{ℏ} t}

Where $n, l, m$ are the quantum numbers, $R_{n l}, Y_{l m}$ are functions in a LUT from these quantum numbers, and $E_{n}$ is the energy of state $n$ .

As an example, in the ground state of hydrogen we get:

Ψ_{100} (r, t) = \frac{1}{\sqrt{π a_{B}}} e^{- \frac{r}{a_{B}} - i \frac{E_{1}}{ℏ} t}

where $a_{B}$ is the Bohr radius.

1.4: Electron Configuration

We use four different quantum numbers to describe the distribution of electrons in orbitals:

principal quantum number ( $n$ )
angular momentum quantum number ( $l$ )
magnetic quantum number ( $m_{l}$ )
spin angular momentum quantum number ( $m_{s}$ )

The configuration is subject to Pauli's exclusion principle

1.4.1: Pauli's Exclusion Principle

It says that:

no two fermions (particles having half-integer spins) in a quantum mechanical system can occupy the same quantum state

In an atom, two electrons in the same orbital can have $n, l, m_{l}$ the same, but they must have different spin angular momentum $m_{s}$ with values $1 / 2$ and $- 1 / 2$

Bosons

This doesn't apply to bosons, or particles with integer spins. They can occupy the same state, due to the nature of the exchange interaction between bosons.

1.4.2: Principal Quantum Number $n$

It denotes the shell or energy level in which the electron can be found, taking values $n = 1, 2, 3, . . .$ The total number of electrons present in a shell is at maximum $2 n^{2}$ .

1.4.3: Orbital Angular Momentum Quantum Number $l$

$l$ describes the subshell. The values orbital angular momentum can have $l = 0, . . ., n - 1$ . They are given by their alphabet labels, starting from $s$ :

l = s, p, d, f, g, . . .

Use the formula:

2 l (2 l + 1)

to find the total number of electrons (hence, use this with the $e_{t o t a l} = 2 n^{2} = 2 l (2 l + 1)$ ).

The $s$ subshell has one orbital
$p$ has 3 orbitals
$d$ has 5
$f$ has 7
...

We describe the electron configuration in increasing subshell, containing the principal quantum number, the subshell, and the number of electrons in the subshell.

1 s^{2} 2 s^{2} 2 p^{6}

For the configuration of Neon (see above), we have:

$n = 1, l = 0$ subshell has 2 electrons ( $1 s^{2}$ )
$n = 2, l = 0$ subshell has 2 electrons ( $2 s^{2}$ )
$n = 2 = l = 1$ subshell has 6 electrons ( $2 p^{6}$ )

Note that sometimes in these we'll contract the configuration to the nearest noble gas, such as in [He] $2 s^{2} 2 p^{6}$ .

2.7: Expectation Values and Variances

Recall that the probability at a point is given by:

P (x, t) = | Ψ (x, t) |^{2}

If we make measurements over a range of an expected value $x$ , then the results may be quite different. But the mean of these measurements can be described by:

⟨ x ⟩ = \int_{- \infty}^{\infty} \hat{x} | Ψ |^{2} d x

where $⟨ x ⟩$ is the expectation value of the position $x$ , and $\hat{x}$ denotes the position operator. Here $⟨ x ⟩$ is the average measurement for $x$ via the quantum state $Ψ$ .

Example

A 6 sided die (fair) has an average (mean) expectation value of 3.5. Calculated via:

⟨ s ⟩ = \sum_{i = 1}^{6} i (1 / 6) = \frac{1 + 2 + \dots + 6}{6} = \frac{21}{6} = 3.5

Note that the 3.5 above doesn't have to make sense with the states actually given, especially if they're finite. As such:

⟨ x ⟩ = \int_{- \infty}^{\infty} \hat{x} Ψ^{*} (x, t) Ψ (x, t) d x

If $Ψ$ is normalized, then this integral should equal 1. Using the linearity of operators from Reading 3 - Electron Configs, Braket Notation, Hilbert Spaces and Sch. Eqtn.#^e742e7:

⟨ x ⟩ = \int_{- \infty}^{\infty} Ψ^{*} (x, t) \hat{x} Ψ (x, t) d x = ⟨ Ψ | \hat{x} | Ψ ⟩

Various measurements will be scattered around $⟨ x ⟩$ . The degree of this scatter is the variance of $x$ and is given by:

σ_{x}^{2} = ⟨ x^{2} ⟩ - {⟨ x ⟩}^{2}

The actual value $σ_{x}$ is the standard deviation of $x$ .

3.3: Hilbert Space

Hilbert Spaces extend math methods of 2D and 3D Euclidian space to finite or infinite dimensions. In QM, we represent the state of a quantum system as a vector (the state vector) in the Hilbert space (called state space). This Hilbert Space $H$ is a complex vector space $C^{n}$ with an inner product. It is defined by:

\forall x \in H, | x | = \sqrt{⟨ x | x ⟩}

A requirement for a Hilbert Space is that it is complete for this norm, or that every Cauchy sequence of elements in $H$ converges to an element in the same $H$ , while the norm of differences between terms approaches $0$ .

A requirement for $Ψ$ in this space is that it is $L^{2}$ , or that $\int | Ψ |^{2} d x$ converges.

Note that $H$ is linear, allowing for superposition.

3.4: Schrodinger Equation

This is a fundamental equation of QM. In classical mechanics, the total energy of a particle is the sum of the kinetic and potential energies:

E = \frac{p^{2}}{2 m} + V (x)

where:

$E$ : energy of the particle
$p$ : momentum of the particle
$m$ : mass of the particle
$V$ : potential energy of the particle, relative to some $x = 0$ .

If the particle is confined to a certain space, the particle requires a minimum energy to escape the potential. If the potential changes, assuming constant energy, then only the momentum changes, thus changing the de Broglie wavelength via:

λ = \frac{h}{p}

where:

$λ$ : de Broglie wavelength
$p$ : the momentum of the particle
$h$ : Planck's constant

Thus giving us the Schrodinger Equation:

i ℏ \frac{\partial Ψ}{\partial t} (x, t) = (- \frac{ℏ^{2}}{2 m} \frac{\partial^{2}}{\partial x^{2}} + V (x, t)) Ψ (x, t)

Since the equation relative to $t$ is a first-order equation, if we know the $t = 0$ part, then we know it for the remaining $t$ . The square of the wavefunction $| Ψ |^{2}$ defines the probability of finding the particle at a given time and place:

| Ψ (x, t) |^{2} = Ψ^{*} (x, t) Ψ (x, t)

this is the probability density. As such:

\int_{- \infty}^{\infty} | Ψ (x, t) |^{2} d x = \int_{- \infty}^{\infty} Ψ^{*} (x, t) Ψ (x, t) d x = 1

Notice that if $Ψ_{1}, Ψ_{2}$ are solutions, then via being a linear operator then a linear combination of these solutions are also solutions:

Ψ = \sum_{i = 1}^{m} a_{i} Ψ_{i}

for all $a_{i} \in C$ . This is the Linear Superposition Principle. Using Dirac's bra-ket notation:

| Ψ ⟩ = \sum_{i = 1}^{m} a_{i} | Ψ_{i} ⟩

where ${| Ψ_{i} ⟩ | Ψ_{2} ⟩, . . .}$ are basis vectors. It's allowed to have $m \to \infty$ here as our basis could be possibly infinite.

The probability of projecting (or measuring) the state $| Ψ ⟩$ into a basis state $| Ψ_{i} ⟩$ is equal to the square of the absolute value of the corresponding probability amplitude, namely $| a_{i} |^{2}$ . The sum of these amplitudes needs to be 1:

\sum_{i = 1}^{n} | a_{i} |^{2} = 1

So then a quantum state can be described as a sum of two or more quantum states, so a quantum system is always in all possible states until a measurement is done. Once that occurs, the wavefunction collapses into one of the more probable states.

3.4.1: Schrodinger's Cat Thought Experiment

Pasted image 20240410223810.png

Given a box with a poison, is the cat alive or dead? Without opening the chamber, the cat is in both states, a superposition of both actually. The definition of the cat state draws from this thought experiment:

| cat ⟩ = \frac{1}{\sqrt{2}} (| dead ⟩ + | alive ⟩)

assuming that both states have a 50% chance of occuring. Notice that despite this chance, we have a $\sqrt{2}$ because each constant is square rooted of the complex probability, namely:

| a_{i} |^{2} = \frac{1}{2} \to a_{i} = \frac{1}{\sqrt{2}}

Lecture Slides: Quantum Superposition

![[Physics CPE 345 Quantum Computing Lecture slides Quantum superposition 240408.pdf#page=19]]

2.2: Complex Numbers

2.5: Linear Vector Spaces

Inner Product

2.6: Using Dirac's Bra-ket Notation

Missed Lecture Slides: Hamiltonian

1.4: Electron Configuration

1.4.1: Pauli's Exclusion Principle

1.4.2: Principal Quantum Number n

1.4.3: Orbital Angular Momentum Quantum Number l

2.7: Expectation Values and Variances

3.3: Hilbert Space

3.4: Schrodinger Equation

3.4.1: Schrodinger's Cat Thought Experiment

Lecture Slides: Quantum Superposition

1.4.2: Principal Quantum Number $n$

1.4.3: Orbital Angular Momentum Quantum Number $l$