Lecture 4 - The Math Behind Quantum Mechanics

Review

We first recall some things from the last lecture.

Difference from $| z |^{2}$ and $z^{2}$

\begin{aligned} | z |^{2} & = (a + b i) (a - b i) = a^{2} + b^{2} \\ = (r e^{i θ}) (r e^{- i θ}) = r^{2} \end{aligned}

Where instead:

\begin{aligned} z^{2} & = z \cdot z = (a + b i) (a + b i) = a^{2} - b^{2} + 2 a b i \neq | z |^{2} \\ = (r e^{i θ}) (r e^{i θ}) = r^{2} e^{i 2 θ} \neq r^{2} \end{aligned}

Note the absence of $i$ 's in $| z |^{2}$ instead of in the $z^{2}$ version.

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) = \hat{H} Ψ (x, t)

here $\hat{H}$ is the Hamiltonian operator. Here:

\hat{H} = i ℏ \frac{\partial}{\partial t}

(make sure to get your OOO's in the word "Schrodinger")

Here $\hat{H}$ denotes the energy of the system. We'll come back to this later.

\hat{H} = - \frac{ℏ}{2 m} \frac{\partial^{2}}{\partial x^{2}} + U (x)

we know that energy is related by $p^{2} / (2 m)$ , hence the similarity to the operations. Here $U (x)$ gives a function that plots the potential at all points $x$ .

3D

Instead we have $x \to \vec{r}$ but the equations and everything are the same. See Reading Week 2 - Intro to Bracket Notation#Missed Lecture Slides Hamiltonian for everything else we missed.

Notice here that there's a time-dependence that allows us to just care at $\vec{r}$ rather than both $\vec{r}$ and $t$ at the same time:

Ψ (\vec{r}, t) = ψ (\vec{r}) e^{- i \frac{E}{ℏ} t}

Hence we really only care about $ψ (\vec{r})$ .

But we needed to know about the quantum numbers. But before that let's look at bra-ket notation.

Diract/Bra-ket Notation

The Dirac (aka bra-ket) notation is a shorthand for writing wave function:

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

becomes:

| n l m ⟩

So:

Ψ_{100} = \frac{1}{\sqrt{π a_{B}}} e^{- \frac{r}{a_{B}} - i \frac{E_{n}}{ℏ} t} \Leftrightarrow | 100 ⟩

and so on.

We have the quantum numbers:

$n$ : Principal quantum number (energy)
$l$ : Electron orbital, the angular momentum quantum number
$m$ : Magnetic quantum number (z-component of electron orbial) angular momentum)

Because of the Heisenburg uncertainty principle, we can only know 2 of the 3 numbers here.

For given $n$ :
$l = 0, 1, . . ., n - 1$
$m = - l, - l + 1, . . ., l - 1, l$

Code for $l$ :
$l = 0$ is s ("sharp")
$l = 1$ is p ("principal")
$l = 2$ is d ("diffuse")
$l = 3$ is f ("funamental")
$l = 4$ is g (alphabetical from here)

If it's lowercase, it's one electron atoms. If it's uppercase, then it has multiple electrons.

Normalization

We know that the probability of our particle anywhere in the region is certain. Hence:

\int_{all space} | Ψ (\vec{r}, t) |^{2} d V = 1

The full version is:

\int_{all space} Ψ^{*} (\vec{r}, t) Ψ (\vec{r}, t) d V = 1

In bra-ket notation:

⟨ Ψ | Ψ ⟩ = 1

The left part is the "bra", and the right part is the "ket". The "bra" is the complex conjugate (and transpose) of the ket. Together, a product makes a "bra-ket", hence the name. However, for our class we really won't need to carry out these integrals.

Using Physical Quantities

Every measurable physical property (position, momentum, energy, ...) has a corresponding quantum mechanical operator $\hat{O}$ . For example, the Hamiltonian operator $\hat{H}$ is an example of the operator for mechanical energy.

Notice that we are working in probabilities, so we have the expectation value, which is the average of measured values of a physical property of a quantum mechanical particle/system:

⟨ \hat{O} ⟩ = \int Ψ^{*} (\vec{r}) \hat{O} Ψ (\vec{r}) d V

For example, the quantum mechanical operator corresponding to energy is $\hat{H}$ , so to find the energy of an electron state in the hydrogen atom:

E_{n} = ⟨ \hat{H} ⟩ = \int {(R_{n l} (r) Y_{l m} (θ, ϕ) e^{- i \frac{E_{n}}{ℏ} t})}^{*} \hat{H} R_{n l} (r) Y_{l m} (θ, ϕ) e^{- i \frac{E_{n}}{ℏ} t}

or:

E_{n} = ⟨ \hat{H} ⟩ = ⟨ n l m | H | n l m ⟩