WKB Approximation

The WKB Approximation

• WKB approximation is used for a particle in a slowly varying potential V(x).
• If the potential varies slowly, particle motion resembles a free particle locally.
• Local wave number is defined as $p(x) = \sqrt{2m(E-V(x))}$.
• Local wavelength is defined as $\lambda(x) = \frac{h}{p(x)}$.
• If $\lambda(x)$ varies slowly ($\frac{d\lambda}{dx} \ll 1$), the solution can be approximated as $\psi(x) \approx A e^{\pm \frac{i}{\hbar} \int p(x') dx'}$.

Justification of WKB Approximation

• The time-independent Schrödinger equation (TISE) is given by $-\frac{\hbar^2}{2m} \psi''(x) + V(x) \psi(x) = E \psi(x)$.
• This simplifies to $\psi''(x) = -\frac{p^2(x)}{\hbar^2} \psi(x)$.
• Let $\psi(x) = e^{\frac{i}{\hbar} S(x)}$, where S(x) is a function used to represent the wave function.
• $\psi'(x) = \frac{i}{\hbar} S'(x) e^{\frac{i}{\hbar} S(x)}$.
• $\psi''(x) = \left[ \frac{i}{\hbar} S''(x) - \frac{1}{\hbar^2} (S'(x))^2 \right] e^{\frac{i}{\hbar} S(x)}$.
• Substituting into the TISE yields $\frac{i}{\hbar} S''(x) - \frac{1}{\hbar^2} (S'(x))^2 = -\frac{p^2(x)}{\hbar^2}$.
• This is rewritten as $i \hbar S''(x) - (S'(x))^2 = -p^2(x)$.
• Expanding $S(x)$ in powers of $\hbar$ gives $S(x) = S_0(x) + \hbar S_1(x) + \hbar^2 S_2(x) + \dots$.
• Substituting the expansion into the equation above yields $i \hbar (S_0'' + \hbar S_1'' + \dots) - (S_0'^2 + 2 \hbar S_0' S_1' + \dots) = -p^2(x)$.
• Equating powers of $\hbar$ yields $(S_0')^2 = p^2(x) \implies S_0' = \pm p(x) \implies S_0(x) = \pm \int p(x') dx'$.
• Also, $i S_0'' - 2 S_0' S_1' = 0 \implies S_1' = \frac{i S_0''}{2 S_0'} = \frac{i (\pm p')}{2 (\pm p)} = \frac{i p'}{2 p}$.
• $S_1(x) = \frac{i}{2} \ln p(x)$.
• $\psi(x)$ is then $e^{\frac{i}{\hbar} S(x)} = e^{\frac{i}{\hbar} (S_0(x) + \hbar S_1(x) + \dots)} = e^{\frac{i}{\hbar} S_0(x)} e^{i S_1(x)} \dots$.
• $\psi(x)$ simplifies to $e^{\pm \frac{i}{\hbar} \int p(x') dx'} e^{-\frac{1}{2} \ln p(x)} = \frac{1}{\sqrt{p(x)}} e^{\pm \frac{i}{\hbar} \int p(x') dx'}$.
• The WKB approximation to the wave function is $\psi(x) = \frac{1}{\sqrt{p(x)}} \left[ A e^{\frac{i}{\hbar} \int p(x') dx'} + B e^{-\frac{i}{\hbar} \int p(x') dx'} \right]$.

Validity of WKB Approximation

• The condition for validity is $\hbar S_1 \ll S_0$ and $\hbar S_1' \ll S_0'$.
• $\hbar \frac{i p'}{2 p} \ll p$, leading to $\frac{\hbar p'}{p^2} \ll 1$.
• Using $p = \frac{h}{\lambda} = \frac{2 \pi \hbar}{\lambda}$, we get $\frac{\lambda'}{2 \pi} \ll 1$.
• WKB approximation is valid when the wavelength varies slowly.

What is Calculus?

• Calculus studies motion and change.
• It is useful for studying motion of planets, changing areas and volumes, tangent lines to curves, and summing infinitely many things.

Main Ideas in Calculus

• Differential calculus deals with the tangent problem.
• Integral calculus deals with the area problem.
• The Fundamental Theorem of Calculus links these two ideas.

Tangent Problem Definition

• Given a function $y=f(x)$, find the equation of the tangent line at the point $(c, f(c))$.
• The slope of the tangent line is given by $m = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$.

Area Problem Definition

• Given a function $y=f(x)$, find the area between the function and the x-axis on the interval $[a, b]$.
• The area is given by $A = \int_a^b f(x) dx$.

Example 1: Tangent Problem

• Problem: Find the slope of the tangent line to $f(x) = x^2$ at $x = 2$.
• Solution: $m = \lim_{x \to 2} \frac{x^2 - 2^2}{x - 2} = \lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2} = \lim_{x \to 2} (x + 2) = 2 + 2 = 4$.

Example 2: Area Problem

• Problem: Find the area under the curve $f(x) = x$ from $x = 0$ to $x = 1$.
• Solution: $A = \frac{1}{2}bh = \frac{1}{2}(1)(1) = \frac{1}{2}$.

Introduction: Digital Transformation in the UH Sector

• Digital transformation means changing a business model and operation via digital tech.
• In the university & higher education (UH) sector, this involves utilizing digital opportunities.
• Goal is to increase quality of research, education, & dissemination, and efficient admin.

Why Digital Transformation is Needed

• Increased student focus as digital tools adapt to individual learning needs.
• Excellent Research is Enabled by data sets & analysis tools that open new research areas.
• Resource efficiency from automating & digitizing administrative tasks frees up core resources.
• Global competition demands digital competence & infrastructure to attract students & researchers.

Overarching Goals of Digital Transformation

• Promote quality in education.
  • Use digital learning resources & platforms to create engaging & flexible experiences.
  • Develop digital assessments with personalized feedback.
  • Strengthen digital fluency by integrating it to study programmes.
• Strengthen research impact.
  • Establish a shared research infrastructure for data sharing & advanced analysis.
  • Promote open research & publishing to increase research availability.
  • Develop incentives for digital cooperation & skills sharing.
• Improve administrative efficiency.
  • Automate manual processes through digitizing & integrating systems.
  • Implement cloud computing solutions to reduce cost & flexibility.
  • Secure data quality & safety for all administrative systems.

Strategies for Digital Transformation

• Competence Development
  • Goal consists of increasing digital competencies to staff & students.
  • Offer digital training programme courses to relevant staff.
  • Integrate digital fluency into study programmes.
  • Mentor schemes & professional networks to share best practice.
• Infrastructure
  • Goal consists of setting up a digital infrastructure that's robust & flexible.
  • Upgrade network capacity & secure access to high-speed internet.
  • Cloud-computing solutions for data processing & storage.
  • Information security & data privacy to all relevant systems.
• Collaboration
  • Goal consists of collaboration and knowledge-sharing among institutions.
  • Establish platforms for shared learning resources and research data.
  • Organise events and conferences to foster networking and digitisation guideline.
  • Frameworks towards standardisation.
• Innovation
  • Experiment/stimulation with digital technology innovation.
  • Establish supports for pilot projects and innovation.
  • Set up labs and 'makerspaces' and facilities for students and staff.
  • Encourage entrepreneurship/commercialization of findings/research.

Implentation of UH Sector Digital Transformation

• Organization
  • Digitalisation committee w/represenatives from applicable fields etc.
  • Coordinate personnel to spearhead this within units.
• Funding
  • Resources for digitalisation in budget allocation.
  • Seek external funding from research facilities etc.
• Evaluation
  • Conduct evaluation of project progression and results etc.
  • Use the subsequent feedback to adjust any applicable strategies.

Conclusion

• Digital transformation is compulsory to ensure UH sector is ready for the future.
• By setting goals, developing effective strategies, investing the necessary resources; a more student-focused, research-intensive and efficient sector can be created.

Channel Capacity

• Channel capacity is the maximum rate for reliable information transmission.
• Reliable communication implies an arbitrarily small probability of error.
• The focus is on the discrete memoryless channel (DMC).

Discrete Memoryless Channel (DMC) Definition

• A DMC includes an input alphabet $\mathcal{X}$, an output alphabet $\mathcal{Y}$, and a conditional probability mass function (PMF) $p(y|x)$ for each $x \in \mathcal{X}$ and $y \in \mathcal{Y}$.
• The memoryless property states that the current output depends only on the current input.

Memoryless Property of DMC

• Mathematically, $p(y_i|x_i, x_1, x_2,..., x_{i-1}, y_1, y_2,..., y_{i-1}) = p(y_i|x_i)$.

Channel Transition Matrix for DMC

• Rows are indexed by inputs $\mathcal{X}$ and columns by outputs $\mathcal{Y}$.
• The entry in row $x$ and column $y$ is the conditional probability $p(y|x)$.
• Examples of DMCs are the Binary Symmetric Channel (BSC) and the Binary Erasure Channel (BEC).

Binary Symmetric Channel (BSC)

• The BSC has input and output alphabets $\mathcal{X} = {0, 1}$ and $\mathcal{Y} = {0, 1}$.
• The channel flips bits with probability $p$.
• The probability $p(y|x) = 1 - p$ if $y = x$, and $p(y|x) = p$ if $y \neq x$.
• The transition matrix for the BSC is $\begin{bmatrix} 1-p & p \ p & 1-p \end{bmatrix}$.

Binary Erasure Channel (BEC)

• The BEC has input alphabet $\mathcal{X} = {0, 1}$ and output alphabet $\mathcal{Y} = {0, 1, e}$.
• The channel transmits a bit correctly with probability $1 - p$ or erases it with probability $p$.
• The probability $p(y|x) = 1 - p$ if $y = x$, $p(y|x) = p$ if $y = e$, and $p(y|x) = 0$ otherwise.
• The transition matrix for the BEC is $\begin{bmatrix} 1-p & p & 0 \ 0 & p & 1-p \end{bmatrix}$.

Channel Capacity Detailed Definition

• It is the maximum mutual information between the input and output, maximized over input distributions $p(x)$: $C = \max_{p(x)} I(X; Y)$.
• $C$ is in bits per channel use.
• $I(X; Y)$ is the mutual information between input $X$ and output $Y$.

Properties of Channel Capacity

• $C \geq 0$ since $I(X; Y) \geq 0$.
• $C \leq \min{\log |\mathcal{X}|, \log |\mathcal{Y}|}$ since $I(X; Y) \leq \min{H(X), H(Y)} \leq \min{\log |\mathcal{X}|, \log |\mathcal{Y}|}$.
• $C$ is a property of the channel alone; it does not depend on how the channel is used. Max rate of infomation transmission.

Calculation of Channel Capacity

• Express mutual information $I(X; Y)$ in terms of $p(x)$ and $p(y|x)$.
• Maximize $I(X; Y)$ over all possible input distributions $p(x)$. This can be done using calculus or numerical optimization techniques.

Example: Channel Capacity of a BSC

• For a BSC with error probability $p$, $C = 1 - H(p) = 1 + p \log_2 p + (1 - p) \log_2 (1 - p)$, where $H(p)$ is the binary entropy function.
  • If $p = 0$, then $C = 1$ bit per channel use.
  • If $p = 0.5$, then $C = 0$ bits per channel use.
  • If $p = 1$, then $C = 1$ bit per channel use.

Example: Channel Capacity of a BEC

• For a BEC with erasure probability $p$, $C = 1 - p$.
  • If $p = 0$, then $C = 1$ bit per channel use.
  • If $p = 1$, then $C = 0$ bits per channel use.

Static Electricity Concepts

• Electric Charge:
  • Two types: positive (proton) and negative (electron); neutron is neutral.
  • Like charges repel; opposites attract; neutral objects are attracted.
• Charge Conservation:
  • Total charge is constant in an isolated system; charge is transferred.
• Charge Polarization:
  • Separation of positive and negative charges in an object.
• Conductors and Insulators:
  • Conductors: Electrons move freely (metals).
  • Insulators: Electrons do not move freely (glass, plastic).

Coulomb's Law

• Equation: $F = k \frac{|q_1 q_2|}{r^2}$ - $F$ is the electric force. - $k = 8.99 \times 10^9 N m^2/C^2$. - $q_1$ and $q_2$ are the magnitudes of the charges. - $r$ is the distance between the charges.
• Key Facts:
  • Vector quantity; force direction connects the charges.
  • Attractive if charges are opposite; repulsive if the same.

Electric Field

• Definition:
  • Force per unit charge; vector quantity.
  • Direction matches the force on a positive test charge.
• Equation: $\mathbf{E} = \frac{\mathbf{F}}{q}$ - $\mathbf{E}$ is the electric field. - $\mathbf{F}$ is the electric force. - $q$ is the charge.
• Point Charge Field: $E = k \frac{|q|}{r^2}$
• Electric Field Lines:
  • Visualize electric fields.
  • Point in the direction of the electric field.
  • Originate on positive charges; end on negative charges.
  • Number of lines relates to charge magnitude.
  • Never cross.
• Electric Dipole:
  • Equal and opposite charges.
  • Electric field is the vector sum of individual fields.
  • Ex: Water molecules.

Electric Potential Energy

• Definition: Energy a charge has due to location in an electric field.
• Change in Energy: $\Delta U = -W$ ($W$ is work done by the electric force).
• Two Point Charges: $U = k \frac{q_1 q_2}{r}$

Electric Potential (Voltage)

• Definition: Electric potential energy per unit charge.
• Quantity Type: Scalar.
• Unit: Volts (V).
• Equation: $V = \frac{U}{q}$
• Point Charge Potential: $V = k \frac{q}{r}$
• Potential Difference: $\Delta V = V_B - V_A = \frac{\Delta U}{q} = - \frac{W}{q}$
• Equipotential Surfaces: Surfaces of constant electric potential.
  • Electric field is perpendicular.
  • No work to move charge.

Capacitance

• Definition: Ability to store electric charge.
• Unit: Farads (F).
• Equation: $C = \frac{Q}{V}$
• Parallel Plate Capacitor: $C = \epsilon_0 \frac{A}{d}$
  • $\epsilon_0 = 8.85 \times 10^{-12} C^2/N m^2$
  • $A$ is the area.
  • $d$ is the distance.
• Stored Energy: $U = \frac{1}{2} C V^2 = \frac{1}{2} Q V = \frac{1}{2} \frac{Q^2}{C}$
• Dielectrics: Insulating materials that increase capacitance when inserted. - Dielectric Constant: $\kappa$ - measure of how much capacitance increases when inserted. So, $C = \kappa C_0$.
• Capacitor Types: Parallel-plate, ceramic, electrolytic, variable.

Fourier Transform Properties Overview

• Fourier Transforms can be manipulated across a variety of properties to adapt to multiple use cases

Fourier Transform Properties

• Linearity:
  • $a f(t) + b g(t) \Leftrightarrow a F(f) + b G(f)$
• Time Shifting:
  • $f(t - t_0) \Leftrightarrow e^{-j\omega t_0} F(\omega)$
• Frequency Shifting:
  • $e^{j\omega_0 t} f(t) \Leftrightarrow F(\omega - \omega_0)$
• Scaling:
  • $f(at) \Leftrightarrow \frac{1}{|a|} F(\frac{\omega}{a})$
• Time Reversal:
  • $f(-t) \Leftrightarrow F(-\omega) = F^*(\omega)$
• Duality:
  • $F(t) \Leftrightarrow 2\pi f(-\omega)$
• Differentiation in Time:
  • $\frac{d}{dt} f(t) \Leftrightarrow j\omega F(\omega)$
  • $\frac{d^n}{dt^n} f(t) \Leftrightarrow (j\omega)^n F(\omega)$
• Integration in Time:
  • $\int_{-\infty}^{t} f(\tau) d\tau \Leftrightarrow \frac{1}{j\omega} F(\omega) + \pi F(0) \delta (\omega)$
• Differentiation in Frequency:
  • $t f(t) \Leftrightarrow j \frac{d}{d\omega} F(\omega)$
  • $t^n f(t) \Leftrightarrow j^n \frac{d^n}{d\omega^n} F(\omega)$
• Convolution:
  • $f(t) * g(t) \Leftrightarrow F(\omega) G(\omega)$
• Multiplication:
  • $f(t) g(t) \Leftrightarrow \frac{1}{2\pi} [F(\omega) * G(\omega)]$
• Parseval's Theorem:
  • $\int_{-\infty}^{\infty} |f(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |F(\omega)|^2 d\omega$

Matrizenmultiplikation (Matrix Multiplication) Definition

• Let $A = (a_{ij})$ be an $m \times n$ matrix and $B = (b_{ij})$ be an $n \times p$ matrix. Then the product $C = A \cdot B$ is an $m \times p$ matrix with entries $c_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}$
• Holds true for $i = 1, \dots, m$ and $j = 1, \dots, p$.

Matrizenmultiplikation (Matrix Multiplication) Example

$A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$, $B = \begin{pmatrix} 5 & 6 \ 7 & 8 \end{pmatrix}$

$C = A \cdot B = \begin{pmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \ 43 & 50 \end{pmatrix}$

(Matrix Multiplication) Properties

• Associativity: $(A \cdot B) \cdot C = A \cdot (B \cdot C)$
• Distributivity: $A \cdot (B + C) = A \cdot B + A \cdot C$ and $(A + B) \cdot C = A \cdot C + B \cdot C$
• In general, not commutative: $A \cdot B \neq B \cdot A$

(Matrix Multiplication) Remarks

• The number of columns of A must match the number of rows of B.
• Matrix multiplication is an important operation in linear algebra and has many applications in various fields such as physics, computer science, and engineering.