Review : Quantum mechanics Jae-hoon Ji Nano Electro-Mechanical Device Lab

Understanding of nature by classical mechanics Newton’s mechanics Solid mechanics Fluid mechanics Governing Equation P = ( AE ⁄ L) δ ⋅ = k δ

Electromagnetism Governing equation Understanding of nature by classical mechanics

Waves behave as particles – Photoelectric effect – Compton effects – Black body radiation Particles behaves as waves – Diffraction – Young’s double-slit experiment However, many experiment results that could not be interpreted by classical model were reported Wave & particle duality

New governing equation - Schrödinger equation Governing equation The role of Newton's laws and conservation of energy in classical mechanicsNewton's lawsconservation of energy

Understanding of Quantum mechanics with the view of EM wave EM wave Wave equation Dispersion relationship Equation for understanding EM wave QM wave Wave equation Dispersion relationship Equation for understanding wave Ψ : Contains all the measurable information about the particle Wave property?  First, we should know Dispersion relationship

Understanding of Quantum mechanics with the view of EM wave Cf) Classical harmonic oscillator – Mass on a spring Undamped oscillator case

The measurable information is obtained by wave function, governing equation. By Mathematics (Differential equation) It is quite important to consider the form of equation and boundary condition Unacceptable forms of ψ must be continuous partial derivatives must also be continuous Mathematics for understanding QM Understanding of Quantum mechanics with the view of EM wave

Mathematics for understanding QM From the view of mathematics, calculating energy of a wave is nothing but calculating the Eigenvalue of equation. Wave function : Contains the measurable information Hamiltonian operator: The operator associated with Energy Eigenvalue of this operation -> Energy 1. The eigenfunctions of Hermitian operators are orthogonal 2. Any wavefunction can be expanded with 3. Coefficients calculated by integration due to orthogonality Cf) Linear algebra Understanding of Quantum mechanics with the view of EM wave

Mathematics for understanding QM Calculation to something you can observe in the laboratory, the "expectation value" of the measurable parameter We can calculate expected value of given wave function. Ex) Understanding of Quantum mechanics with the view of EM wave

QM wave 1D - Infinite potential wall EM wave 1D – Perfect conductor wall Understanding of Quantum mechanics with the view of EM wave r=-1(Reflection coefficient) ? ?

QM wave 1D - finite potential wall EM wave 1D – Dielectric k1k2 Wave is continuous 1 st derivative of wave is continuous Understanding of Quantum mechanics with the view of EM wave

QM wave 1D - finite potential wall Previous case Understanding of Quantum mechanics with the view of EM wave Previous case

QM wave 1D - Infinite potential well EM wave Understanding of Quantum mechanics with the view of EM wave 1. Equation 2. Wave function & B.C 1. Equation 2. Wave function & B.C k

Understanding of Quantum mechanics with the view of EM wave EM wave 1. Equation 2. Wave function & B.C

Understanding of Quantum mechanics with the view of EM wave QM wave 1. Equation 2. Wave function & B.C

Understanding of Quantum mechanics with the view of EM wave κ κ κ κ κ κ L L L L L L L κ κ

QM wave How about coupled potential well? Understanding of Quantum mechanics with the view of EM wave 1. Equation 1. Individually As a matrix

1. Equation 1. Individually 2. Close together 2.a Close look at [S] 2.b Close look at [H]

We can find Eigen state and eigenfunction

Crystal structure Understanding of Quantum mechanics with the view of EM wave Bloch wave Isosurface of a Bloch wave in silicon lattice Wavefunction for a particle in a periodically-repeating environment, most commonly an electron in a crystal Multiply a plane wave by a periodic function 1. Under the periodic potential 2. The eigenstates ψ of the Hamitonian

Understanding of Quantum mechanics with the view of EM wave Crystal