Lecture 1: Introduction - Physical Chemistry II - SIC2020 PDF

# Lecture 1: Introduction ## Shameer Hisham ## SIC2020 Physical Chemistry II ## Course Information - **Quantum Chemistry** - 14 hours - **Lecturer**: - Shameer Hisham - B023 (office) / B005 (lab) - 03-7967 4081 (office) / 6787 (lab) - [email protected] - **Sharing with**:...

# Lecture 1: Introduction ## Shameer Hisham ## SIC2020 Physical Chemistry II ## Course Information - **Quantum Chemistry** - 14 hours - **Lecturer**: - Shameer Hisham - B023 (office) / B005 (lab) - 03-7967 4081 (office) / 6787 (lab) - [email protected] - **Sharing with**: - Dr. Nor Mas Mira (Chemical Kinetics) - 10 hours - Assoc. Prof. Dr. Noor Idayu (Chemical Thermodynamics) - 10 hours - Assoc. Prof. Dr. H. N. M. Ekramul Mahmud (Macromolecules) - 8 hours ## Assessment and Grading - **Assessments**: - Final examination: 60% (20% for my part) - Continuous assessment: 40% (10% for my part) - Only one assignment - Individual assignment (Excel, Gaussian 09W and/or LATEX) ## The Schrödinger Equation ### Time-Dependent 1-D Schrödinger Equation $ Ĥψ(x,t) = ih \frac{∂}{∂t}ψ(x,t) $ (1) - Using the separation of variables method to solve this PDE, we can obtain the time-independent Schrödinger equation: ### Time-Independent 1-D Schrödinger Equation $Ĥψ(x) = Εψ(x) $ (2) - It is this form that you are familiar with, and only this form is considered during the course of our lectures. ## The Probability Density ### The Born Interpretation The square of a wavefunction, $ψ^²$ is directly proportional to the probability of finding particles (e.g. electrons) at a specified space. ### In 1-D space: $P = \int_{x0}^{x1} |ψ|^2 dx = \int_{x0}^{x1}ψ^*ψ dx$ - The sum over all the space is: $\int_{-\infty}^{+\infty}ψ^*ψ dx = 1$ ## The Probability Density When we integrate this wavefunction from 0 to L, we find that the definite integral equals 0. $ψ^2 = \sqrt{\frac{2}{L}}sin(\frac{2πx}{L})$ ## The Probability Density Now, when we integrate this probability density function, $|ψ|^2$ from 0 to L, we find that the definite integral equals 1. $∫|ψ^2| = ∫^L_0(\sqrt{\frac{2}{L}}sin(\frac{2πx}{L}))^2 dx = 1$ ## The Schrödinger Equation in 3-D Space - In 3-D space, the time-independent Schrödinger equation in the Cartesian coordinate system is: $-\frac{ħ^2}{2m}∇^2(ψ) + Vψ = Εψ$ (3) where $∇^2 = \frac{∂^2}{∂x^2} + \frac{∂^2}{∂y^2} + \frac{∂^2}{∂z^2}$ - This wavefunction is normalised if and only if $∫_{-∞}^{+∞}∫_{-∞}^{+∞}∫_{-∞}^{+∞}ψ^*ψ dx dy dz = 1$ (4) ## Spherical Polar Coordinates - The Schrödinger equation for a hydrogen atom in Cartesian coordinates is: $\frac{∂^2}{∂x^2} + \frac{∂^2}{∂y^2} + \frac{∂^2}{∂z^2}ψ + \frac{ħ^2}{2m}(\frac{e^2}{4πε_0r}) = Eψ$ - We see that this PDE cannot be separated into three separate PDEs because $r = \sqrt{x^2 + y^2 + z^2}$. Solving this PDE would be a horrifying nightmare! ## Spherical Polar Coordinates - Thus, it would be much more convenient if we work with spherical polar coordinates (r, θ, φ), where $\frac{1}{r^2} \frac{∂}{∂r}(r²\frac{∂ψ}{∂r}) + \frac{1}{r²sin(θ)}\frac{∂}{∂θ}(sin(θ)\frac{∂ψ}{∂θ})+\frac{1}{r²sin²(θ)}\frac{∂²ψ}{∂φ^2} + \frac{2m}{ħ²}(E-\frac{e²}{4πε_0r})ψ = 0$ (5) - Correspondingly, the probability density in spherical polar coordinates is: $\int_0^{∞} \int_0^{π} \int_0^{2π}ψ^*ψ r² dr sin(θ) dθ dφ= 1$ (6) and $dτ = dx dy dz = r² dr sin(θ) dθ dφ$ ## Spherical Polar Coordinates - A diagram of spherical polar coordinates is shown. - The coordinate system in spherical polar coordinates is given by r, θ, φ, where r is the distance from the origin, θ is the angle with the z-axis and φ is the angle in the x-y plane. ## Orthonormality - A wavefunction is normalised if it satisfies the following condition: $\int_{-∞}^{+∞}ψ^*ψ dx = 1 $ - Two wavefunctions are orthogonal if they satisfy the following condition: $\int_{-∞}^{+∞}ψ^*_mψ_n dx = 0$ if $m≠n$ - Orthonormal wavefunctions satisfy the following condition: $\int_{-∞}^{+∞}ψ^*_mψ_n dx = \begin{cases} 0 & \text{if } m≠n \\ 1 & \text{if } m = n \end{cases}$ ## Particle in a Box - A diagram of a particle in a box is shown. - The wavefunction for a particle in a box is given by $ψ_n = \sqrt{\frac{2}{L}}sin(\frac{nπx}{L})$ (7) ### 1-D Schrödinger equation - The energy of a particle is limited to: $E = \frac{n²ħ²}{8mL²}$ , n = 1, 2,... (8) - Remember, energy of a quantum particle is QUANTISED! ## Thank You!