Quantum Mechanics: Particle in a Box

Study Notes

Particle in a One-Dimensional Box

In a one-dimensional box, potential energy outside the boundaries (0 and a) is infinite while inside is zero.
Potential energy function: ( V(x) = 0 ) for ( 0 < x < a ), ( V(x) = \infty ) for ( x \leq 0 ) and ( x \geq a ).

3D Schrödinger Equation

The general form of the 3D Schrödinger equation incorporates kinetic and potential energy terms: [ \frac{-ħ^2}{2m} \nabla^2 \psi + (E - V)\psi = 0 ]
For a particle in a box, the potential energy ( V ) is defined similarly as above.

Wave Function and Boundary Conditions

The second derivative of the wave function ( \psi ) gives rise to a quantization condition: [ \frac{d^2 \psi}{dx^2} + k^2 \psi = 0 ]
The wave function can be expressed as: [ \psi = A \sin(kx) + B \cos(kx) ]

Boundary Condition Analysis

At boundaries ( x = 0 ) and ( x = a ), the wave function must equal zero.
Setting ( \psi(0) = 0 ) leads to ( B = 0 ).
At ( x = a ), ( \sin(ka) = 0 ) implies ( ka = n\pi ), where ( n = 1, 2, 3, \ldots ), leading to ( k = \frac{n\pi}{a} ).

Energy Levels

Energy levels are quantized and given by: [ E_n = \frac{n^2 h^2}{8ma^2} ]
Corresponding wave functions are: [ \psi_n = \sqrt{\frac{2}{a}} \sin\left(\frac{n\pi x}{a}\right) ]

Energy Differences

Energy difference between states can be calculated as: [ \Delta E = E_2 - E_1 = \frac{3h^2}{8ma^2} ]
Successive differences highlight the increasing energy gaps: [ E_3 - E_2 = \frac{5h^2}{8ma^2} ]

Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO)

HOMO corresponds to ( n ) and LUMO to ( n + 1 ).
Energy for ( n + 1 ) state is expressed as: [ E = \frac{(n+1)^2 h^2}{8ma^2} ]

Conjugated Molecules

Defined by overlapping p-orbitals, affecting the molecular geometry.
Example bond lengths: C-C = 154 pm; C=C = 134 pm.

Molecular Orbital (MO) Diagram

Energy levels can be represented in diagrams indicating the energy states of molecular electrons.
Configuration in terms of ( \psi_n ) captures the energy and wave function behavior within the system.

Atomic Orbitals

Atomic orbitals define 3D spaces around the nucleus that represent probable locations of electrons.

Schrödinger Equation for Hydrogen Atom

The modified Schrödinger equation for hydrogen considers kinetic and potential energies: [ \frac{d^2 \psi}{dx^2} + \frac{8\pi^2 \mu}{h^2} (E - V) \psi = 0 ]
The effective mass appears in the equation, representing the influence of both electron and nucleus.

Electron Motions in Hydrogen Atom

The motions are categorized as revolution around the nucleus and translational motion.

Example Calculation for Energy Transition

Energy required to promote an electron from HOMO to LUMO for linear hexa-1,3,5-triyne can be computed using the derived energy equations for respective states.

Description

Test your understanding of the particle in a one-dimensional box concept in quantum mechanics. Explore potential energy behavior inside and outside the box, as well as the 3D Schrödinger equation. This quiz will enhance your grasp of key quantum principles and equations.