Quantum Mechanics Lecture Notes PDF

# Quantum Mechanics ## DeBroglie's Hypothesis - λ = h/p - λ is the wavelength of a particle - p is the momentum of a particle - deBroglie's hypothesis states that all matter has wave-like properties ## Wave Function - ψ = A cos(kx) - A is the amplitude of the wave - k is the wave number = 2π/λ -...

# Quantum Mechanics ## DeBroglie's Hypothesis - λ = h/p - λ is the wavelength of a particle - p is the momentum of a particle - deBroglie's hypothesis states that all matter has wave-like properties ## Wave Function - ψ = A cos(kx) - A is the amplitude of the wave - k is the wave number = 2π/λ - kħ = p - ħ is the reduced Planck constant - kħ = 2πħ /λ = h/λ = p - Wave vector; k = 2π/λ ## Heisenberg's Uncertainty Principle - Δx Δp >= ħ/2 - Δx is the uncertainty in position - Δp is the uncertainty in momentum - Heisenberg's uncertainty principle states that it is impossible to know both the position and momentum of a particle with perfect accuracy. ## Quantum Mechanics: Postulates ### Postulate 1 - The state of a system is fully specified by the **state function** or **wave function**: ψ(x, y,... t) - ψ(x, y,... t) is a function of space and time - The probability of finding a particle in a small volume dxdydz is: |ψ(x, y, z, t)|² dxdydz - Probability density = |ψ(x, y, z, t)|² - Total probability = ∫ |ψ(x, y, z, t)|² dxdydz = 1 - In order for the wave function to be physically meaningful, it must be **square integrable**. ### Postulate 2 - Corresponding to every observable, there is one **linear operator**. - **Linear operator:** An operator A is linear if: A(c₁f₁ + c₂f₂) = c₁Af₁ + c₂Af₂ - **Examples of linear operators:** - Position: x - Momentum: p = -iħ(d/dx) - Kinetic energy: K.E. =-ħ²/2m(d²/dx²) - Potential energy: V(x) ### Postulate 3 - Measurement of an observable gives one of the **eigenvalues** of the corresponding **operator**. - An **eigenvalue equation** is an equation of the form: Aψ(x, y, z, t) = a ψ(x, y, z, t) - In this equation, **a** is the eigenvalue, and **ψ(x, y, z, t)** is the eigenfunction. ### Postulate 4 - The **wave function** obeys the **time-dependent Schrödinger equation**: iħ(∂/∂t)ψ(x, y,... t) = Ĥψ(x, y,... t) - This equation describes the evolution of the wave function over time. ### Consequences of Postulate 4: - If the potential energy V(x) is time-independent: - **Stationary state**: ψ(x, t) = ψ(x)e^(-iEt/ħ) - **Standing waves**: ψ(x, t) = ψ(x)e^(-iEt/ħ) ## Linear Superposition - A general wave function in space can be written as a linear combination of the eigenfunctions: ψ(x) = Σc₁ψ₁(x) - The coefficients c₁ are complex numbers - The eigenfunctions are orthonormal: ∫ψᵢ(x)ψⱼ(x)dx = δᵢⱼ ## Time-Dependent Schrodinger Equation - The time-dependent Schrodinger equation (TDSE) describes the time evolution of a quantum system. - The solution to the time-dependent Schrodinger equation is the wave function ψ(x, t) - In the case of a time-independent potential: - The wave function can be separated into a product of a time-independent part ψ(x) and a time-dependent part T(t): ψ(x, t) = ψ(x)T(t). - The time-independent part of the wave function ψ(x) is called the **stationary state**. ## Simple Harmonic Oscillator (SHO) - The simple harmonic oscillator (SHO) is a fundamental system in quantum mechanics - The Hamiltonian of the SHO is: - H = -ħ²/2m(d²/dx²) + 1/2kx² - The energy eigenvalues of the SHO are: - E = (n + 1/2) ħω - The energy eigenstates of the SHO can be expressed as a combination of the Hermite polynomials and the Gaussian function. ## Expectation Value - The expectation value of an observable A is given by: - <A> = ∫ψ*(x)Aψ(x)dx ## Conjugated Systems - Conjugated systems are molecules that have alternating single and double bonds. - The π electrons in these systems are delocalized over the entire molecule. - The potential energy of the π electrons in a conjugated system can be approximated by a box potential model. - The energy levels of the π electrons are quantized. - The number of energy levels is equal to the number of π electrons. - In the limit of an infinite box, the energy levels become continuous. This provides a good approximation for large conjugated systems. **Please note that I have excluded images from this document. You can refer to the original image for the illustrations.**