Particles are traditionally treated as points that can be found at some position in 3-dimensional space for a given time.
However when looking closer at the makeup of a particle we find the makeup consists of a bit more than a mere point.
We'll consider the makeup of a particle to be some unknown function \( \psi \) that is a function of its position in space: \( \psi = \psi(x,y,z) \).
More on this later.
Leaving the makeup ambiguous for the time being, we can say that there are properties of the particle that can be observed from it. These are called, 'observables.'
Operators
Operator \( {\hat A}(\psi) \) has expected value \( \langle A \rangle_\psi = {{\langle \psi, {\hat A}(\psi) \rangle} \over {\langle \psi, \psi \rangle}} \).
For continuous real functions we have an inner product \( \langle f, g \rangle = \int_{\mathbb{R}^3} (f^* \cdot g) dx \)
and our expected value of a normalized function is \( \langle A \rangle_\psi = \int_{\mathbb{R}^3} (\psi^* \cdot A \psi) dx \).
The density of the particle is given by \( \rho = |\psi|^2 = \langle \psi, \psi \rangle \).
If \( {\langle \psi, \psi \rangle} = 1 \) then the function \( \psi \) is said to be normalized
and the expected value simplifies to \( \langle A \rangle_\psi = \langle \psi, {\hat A}(\psi) \rangle \).
Once the function is normalized, the probability of finding the particle in a certain location is \( \int_S |\psi|^2 dx \).
Observables
The observation of the position is \( {\hat x}(\psi) = x \cdot \psi \).
For the i'th dimension, \( {\hat x}^i(\psi) = x^i \cdot \psi \).
In other words, \( {\hat x}(\psi) = x \cdot \psi \), \( {\hat y}(\psi) = y \cdot \psi \), and \( {\hat z}(\psi) = z \cdot \psi \).
The expected value of the measurement of a particle's position is denoted \( \langle x \rangle \) and calculated as \( \langle x \rangle = \int x |\psi|^2 dx \).
The observation of the momentum along the x-direction, called the momentum operator, is denoted \( {\hat p_x}(\psi) = -i \hbar \nabla_x \psi \) for \( \hbar = {h \over {2 \pi}} \) and for h representing Planck length.
The momentum observation along the i'th dimension is \( {\hat p}_i(\psi) = -i \hbar \nabla_i \psi \).
Shorthand, \( {\hat p}_i = -i \hbar \nabla_i \).
To compute the momentum along a normal: \( n^i {\hat p}_i = -i \hbar n^i \nabla_i \).
In classical mechanics, the total energy in a system is H (for "Hamiltonian") and is defined as \( H = T + V \) for kinetic energy T and potential energy V.
Kinetic energy is defined as \(T = {1 \over 2} m v^2 \).
Momentum is defined as \( p = mv \).
From there we can calculate
\( T = {1 \over 2} m ({p \over m})^2 = {p^2 \over {2 m}} \).
Potential energy, V, is left undefined.
In quantum mechanics things work the same, but with operators replacing variables. Total energy is \( {\hat H} = {\hat T} + {\hat V} \).
Kinetic energy is defined as
\({\hat T} = {1 \over {2 m}} ({\hat p}_i)^2 \).
\({\hat T} = {1 \over {2 m}} ({\hat p}_x^2 + {\hat p}_y^2 + {\hat p}_z^2) \).
\({\hat T} = {1 \over {2 m}} ((-i \hbar \nabla_x)^2 + (-i \hbar \nabla_y)^2 + (-i \hbar \nabla_z)^2) \).
\({\hat T} = -{{\hbar^2} \over {2 m}} (\nabla_x^2 + \nabla_y^2 + \nabla_z^2) \).
\({\hat T} = -{{\hbar^2} \over {2 m}} (\nabla_i)^2 \).
Potential energy is considered a multiplicative operator that is a function of position alone.
From this our total energy operator becomes \( {\hat H} = {\hat T} + {\hat V} = -{{\hbar^2} \over {2 m}} \nabla_i^2 + {\hat V} \).
Angular Momentum.
\( J^k = \epsilon_{ijk} x^i p_j \). I'll reconcile index gymnastics later.
For example, \( J^x = {\hat y} {\hat p}_z - {\hat z} {\hat p}_y \).
To apply the momentum operator, consider the wavefunction \( \psi = e^{i(a_i x^i)/\hbar} \). Then
\( {\hat p}_j \psi = -i \hbar \nabla_j e^{i(a_i x^i)/\hbar} \)
\( {\hat p}_j \psi = -i \hbar \cdot i / \hbar a_j e^{i(a_i x^i)/\hbar} \)
\( {\hat p}_j \psi = a_j \psi \)
\( {\hat p}_j = a_j \)
This says that our particular choice of \( \psi = e^{i(a_i x^i)/\hbar} \) is an eigenvector of \( {\hat p}_j \), with associated eigenvalue \( a_j \).
In such a state the kinetic energy measured is \( {\hat T} = {{\hat p}_i^2 \over {2m}} \).
Hydrogen Atom
Associated Legendre Function: \( P^m_l(t) = {{(1 - t^2)^{m/2}} \over {2^l l!}} {{d^{l+m}} \over {dt^{l+m}}} (t^2 - 1)^l \)
potential: \( V(r) = - {{Z e^2} \over {r}} \)
\(Z = {1 \over {4 \pi \epsilon_0}} \)
Bohr radius: \( a_0 = 0.527 \times 10^{-10} \)
Radial Solution
\( n = 1, 2, 3, ... \)
\( l = 0, 1, 2, ..., n - 1 \)
energy eigenvalues: \( E_n = - {{\mu Z^2 e^4} \over {2 \hbar^2 n^2}} \)
\( \alpha_n = {1 \over {n a_0}} \)
\( c_j = \left [ {{2 \alpha_n (j-n)} \over {j(j+1) - l(l+1)}} \right ] c_{j-1} \)
\( p_{nl}(r) = c_l r^l + c_{l+1} r^{l+1} + ... + c_{n-1} r^{n-1} \)
radial solution: \( R_{nl}(r) = p_{nl}(r) e^{-\alpha_n r} \)
Hydrogen Atom
\( n = 1, 2, 3, ... \)
\( l = 0, 1, 2, ..., n - 1 \)
\( m = -l, ..., -1, 0, 1, ..., l \)
wavefunction: \( \psi_{nlm}(r,\theta,\phi) = R_{nl}(r) P^m_l(cos(\theta)) e^{im\phi} \)
