Presentation on theme: "An Electron Trapped in A Potential Well Probability densities for an infinite well Solve Schrödinger equation outside the well."— Presentation transcript:
An Electron Trapped in A Potential Well Probability densities for an infinite well Solve Schrödinger equation outside the well
inside the well An Electron Trapped in A Potential Well The general solution is By the boundary conditions
An Electron Trapped in A Potential Well Normalization For odd wave function For even wave function probability
inside the well An Electron Trapped in A Potential Well energy distribution
Barrier Tunneling Dividing the space into three parts: I) to the left of the barrier II) within the barrier; and III) to the right of the barrier The conditions for wave functions at the boundary are continuity.
Wave (matter wave) Uncertainty Principle Wave Function Schrödinger’s Equation probability Free Electrons Hydrogen atom de Broglie relation, de Broglie wavelength. Potential Well Particle The Nature of Matter Energy quantization Probability Barrier Tunneling STM
Schrödinger’s Equation =0.529ÅBohr radius ground state (n=1) n=2
ground state Energy quantization Principle quantum number Schrödinger’s Equation
The Uncertainty Principle The angular momentum-angle Uncertainty Relationship
Schrödinger’s Equation Angular Momentum of Electrons in Atoms Angular momentum quantum number Angular momentum quantization
l 0 1 2 3 4 5 code s p d f g h Using this labeling, we can express 1s for ground state (n=1, l=0). The first excited state has two designations: 2s (n=2, l=0) and 2p (n=2, l=1). Angular momentum quantum number l=0, 1, 2, …, n-1. Ex. n=1, l=0 for s state n=2, l=0, 1. for l=0, s state and l=1, p state
Space quantization m l =-l, -(l-1), …-1, 0, 1, …,(l-1), l magnetic quantum number
n Principle quantum number l Angular momentum quantum number m l =-l, -(l-1), …-1, 0, 1, …,(l-1), l m l magnetic quantum number
(n=1, l=0) The Ground State ground state (n=1) m l =0, is spherically symmetric.
The 2s State (n=2, l=0) m l =0, is spherically symmetric.
An Excited State Of The Hydrogen Atom The 2p State (n=2, l=1) m l =-1, 0, +1 is not spherically symmetric.
Electron Spin Pauli pointed out the need for a 4 th quantum number in 1924—Spin quantum number and l Angular momentum quantum number Spin magnetic quantum number s Spin momentum quantum number
The States of Atomic Hydrogen The assembly of all hydrogen-atom states with the same principal quantum number n are said to form a shell. The collection of all states with the same value of the orbital angular momentum quantum number l is called a subshell. For a certain angular momentum l, there are 2l+1 states, and consider spin there are 2(2l+1) states. m l =-l, -(l-1), …-1, 0, 1, …,(l-1), l
states Example m l =-l, -(l-1), …-1, 0, 1, …,(l-1), l states
The X-Ray Spectrum of Atoms The Characteristic X-ray Spectrum
Atomic Magnetism How to study the angular momentum properties of the atom ? L is the orbital angular momentum vector of the electron. The component of z direction L z =m l ħ, m l =-l, -(l-1),...-1, 0, +1,..., +l
Atomic Magnetism L z =m l ħ, m l =-l, -(l-1),...-1, 0, +1,..., +l It can be expressed by Bohr magnetron µ B We can express the magnetic dipole moment in terms of the Bohr magnetron
If we were to place an atom having a magnetic dipole moment in a magnetic field, which we assume is in the z direction, the energy associated with the interaction between the atom and the magnetic field is That is, atoms with different values of m l have different energies in the field, which provides a way to determine their orbital angular momentum. Atomic Magnetism