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MindMap Gallery College Physics Quantum Physics Fundamentals
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College Physics Quantum Physics Fundamentals
Quantum Mechanics College Physics, including wave-particle duality, Wave functions, typical quantum phenomena of the Schrödinger equation, operators of mechanical quantities representing quantum measurements, atomic structures, etc.
Edited at 2024-01-19 15:57:19
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College Physics Quantum Physics Fundamentals
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Fundamentals of Quantum Physics
wave-particle duality
Blackbody Radiation Problem Energy Quon Hypothesis
Experimental rules of black body radiation
When the temperature of an object increases, it emits heat to its surroundings, which is called thermal radiation
Monochromatic radiance
The integration of monochromatic radiance over wavelength gives the radiant power per unit area of the object surface, which is called radiance.
monochromatic absorptivity
The absorptivity of an object is the ratio of the energy absorbed per unit area of the surface of the object to the incident energy
Blackbody monochromatic absorptivity is 1
Kirchhoff's Law
A black body has the strongest radiation and strongest absorption at the same temperature.
blackbody energy spectral density
The distribution of radiation field energy relative to wavelength in unit volume is only related to temperature
law
For a certain temperature, the shape of the curve is certain, regardless of material, shape or size.
Stefan Boltzmann's law
Wien's displacement law
The difficulty of classical theory in explaining the experimental rules of black body radiation
Wien's formula
short wave area
Rayleigh's formula
Long wave area
Very short waves will diverge
UV disaster
Planck's formula Energy quantum hypothesis
For electromagnetic radiation of a certain frequency, objects can only emit or absorb electromagnetic waves in units of hv.
Objects emit or absorb electromagnetic waves only in a quantum manner
Photon wave-particle duality of light
Einstein's Quantum Hypothesis
The electromagnetic field itself is also quantized. The electromagnetic field is composed of a limited number of energy quanta, each of which is limited to a small volume of space. These energy are no longer dispersed in motion and can only be absorbed or generated entirely. These energy quanta are referred to as Photon
photoelectric effect phenomenon
Optoelectronics
Flying towards the anode under the action of accelerating electric field to form photocurrent
For a cathode made of a certain metal material, photoelectrons will be emitted only when the irradiation light frequency is greater than the cutoff frequency v0.
When the irradiation light frequency is constant, the photocurrent increases with the increase of the forward acceleration voltage and tends to the saturation value, and the saturation current is proportional to the incident light intensity.
When the accelerating voltage is 0, there are still some electrons with high initial kinetic energy reaching the anode. Adding a reverse voltage between the two poles reduces the photocurrent to a cut-off voltage of 0.
The cut-off voltage or initial kinetic energy of photoelectrons is only linearly related to the frequency of light
Photocurrent and light irradiation occur almost simultaneously
Light quantum explanation of the law of photoelectric effect
w0 is called the work function
Light intensity only affects the number of electrons produced
photon energy and momentum
Compton effect and light quantum explanation of Compton effect
The phenomenon that the wavelength of electromagnetic waves changes after being scattered by matter is called the Compton effect
Wave-particle duality of light
Light has the dual nature of waves and particles
Bohr's theory of atomic structure
nucleated model of atom
Rutherford proposed
Experimental rules of hydrogen atom spectrum
Baltimore formula
Rydberg formula
Bohr's theory of atomic structure
Steady state conditions
Frequency condition
Stationary state quantization conditions
radius
When n=1, it is Bohr radius
Limitations of Bohr's theory
Powerless except for hydrogen atoms
It cannot be explained why quantization is required
Wave nature of physical particles Matter waves
De Broglie hypothesis of wave nature of physical particles
The two equations are de Broglie relations
de Broglie wavelength
Experimental proof of particle wave properties
Explanation of matter waves Probability waves
Matter waves are not fluctuations of actual physical quantities, but probability waves that describe the spatial distribution of particles. They guide the movement of particles and determine the probability of particles appearing at each point.
How electrons pass through double slits
wave function
uncertainty relation
Coordinate momentum uncertainty relationship
Heisenberg uncertainty principle
Energy time uncertainty relationship
Examples of application of uncertainty relations
Description of the motion state of microscopic particles Wave function
Freely moving single particle wave function
Dirac symbol
The first basic assumption of quantum mechanics: the motion state of microscopic particles is described by wave functions
Normalization of the wave function
normalization condition
Satisfying this condition is called a normalized wave function
Multi-particle system wave function
Normalization conditions for wave functions
square integrable
Single value bounded
Continuously differentiable
The principle of quantum state superposition
Electron diffraction experiment through metal polycrystalline film
The state superposition principle of quantum mechanics
If a quantum system can be in the state described by wave functions 1 and 2, it can also be in its linear superposition state.
The second postulate of quantum mechanics
Normalization of plane wave wave function Momentum value frequency
Delta function normalization method for plane wave function
Box normalization method for plane wave wave functions
Probability of momentum distribution in a given state
Coordinate representation and momentum representation
Wave function meaning
Schrödinger equation typical quantum phenomena
Schrödinger equation
Schrödinger equation
condition
The wave function satisfies the principle of linear superposition
Parameters related to specific states such as momentum and energy cannot be included in the equation.
The classical limit can be transitioned to the equations of classical mechanics
Wave equation of non-relativistic free particles
in a powerful field
Non-relativistic quantum mechanical equations
Energy operator Momentum operator Hamiltonian operator
The state vector of an isolated quantum system changes with time according to the Schrödinger equation
energy operator
Momentum operator
kinetic energy operator
Potential energy operator
Hamiltonian operator
The Schrödinger equation can be written as
Discussion about Schrödinger equation
The Newton's equation language is Laplace's determinative, while the Schrödinger equation's values of physical quantities are non-deterministic, probabilistic, and statistical.
The Schrödinger equation is non-relativistic
Relativistic quantum mechanical equations
Klein-Golden equation
Stationary Schrödinger equation
Stationary Schrödinger equation
If the potential function of the system has nothing to do with time and the Hamiltonian of the system is only a function of the spatial coordinates and has nothing to do with time, such a system is called a stationary system.
Variables can be separated
stationary equation
Therefore, the probability density of the system in the stationary state has nothing to do with the event.
stationary wave function
Eigenvalue problem
Eigenvalue equation
The constant A is the eigenvalue
There will only be specific eigenvalues
For different problems, the eigenvalue E of the Hamiltonian operator may take a continuous value (continuous spectrum) or a discrete discrete value (discrete spectrum). The possible values of E are called energy levels. For a certain energy level, if If there is only one linearly independent eigenfunction, the energy level is non-degenerate. The number of linearly independent eigenfunctions at the same energy level is called the degeneracy of the energy level.
The motion of particles in a one-dimensional infinitely deep potential well
One-dimensional infinite deep potential well problem
Motion characteristics of particles in infinitely deep potential wells
Particle energy can only take discrete values, n is called the energy quantum number
The lowest energy of the particle is not equal to zero, which is called the ground state
Matter waves form standing wave distribution in the well
Wave functions belonging to different energy eigenvalues are orthogonal to each other
All eigenfunctions are complete
One-dimensional linear resonator
One-dimensional harmonic oscillator
The movement of any particle near the equilibrium point can be approximated by simple harmonic motion
quantum oscillator
Satisfies the stationary Schrödinger equation
The solution contains a Hermitian polynomial
Basic characteristics of quantum oscillators
Energy levels are discrete spectrum
The ground state energy of the quantum oscillator (n=0) is called zero point energy
The probability density distribution of quantum harmonic oscillators bears no resemblance to classical oscillators
At a point where the probability density of the classical oscillator is not equal to zero, the quantum oscillator can be zero and can reach the area that the classical oscillator cannot reach.
barrier penetration
One-dimensional barrier
barrier penetration
Transmission coefficient
Reflection coefficient
In general, the reflection coefficient is not zero
Transmission coefficient
The transmission coefficient decreases exponentially as the barrier widens and heightens and the particle mass increases.
Reflection coefficient
The phenomenon that particles with energy smaller than the barrier height still penetrate the barrier is called tunneling effect.
scanning tunneling microscope
Experimental Proof and Technical Application of Quantum Tunneling Effect
Operator representation of mechanical quantities Quantum measurement
Linear Hermitian operator
Linear Hermitian operator
The fourth postulate of quantum mechanics: Mechanical quantities in quantum mechanics are represented by linear Hermitian operators
If the two are equal, the operator is a Hermitian operator.
Eigenvalues and eigenfunctions of linear Hermitian operators
The eigenvalues are all real numbers
Eigenfunctions with different eigenvalues are orthogonal
The linear Hermitian operator eigenfunction serves as a basis vector to form a complete vector space.
Mechanical quantities are represented by linear Hermitian operators
The average value of coordinates and momentum at a given state
Mechanical quantities are represented by Hermitian operators
The construction of mechanical quantity operators
Angular momentum operator
Commutation relationship of operators Physical meaning of commutation relationship
Yizi
Mechanical quantity operators generally do not satisfy exchangeability (commutability)
The physical meaning of operator commutation
If equal to zero, it is commutable
The number of operators included in the complete concentration of mechanical quantities is equal to the number of degrees of freedom of the system
General form of uncertainty relation between two non-commutative mechanical quantity operators
Angular momentum operator Angular momentum operator eigenvalues and eigenfunctions
Representation of angular momentum operator in spherical coordinate system
If this commutation relationship is satisfied, it is the angular momentum operator
Angular momentum operator eigenvalues and eigenfunctions
Spherical harmonics
l is called the orbital (angular momentum) quantum number
m is the magnetic quantum number
Angular momentum quantization
Orbital range is quantized
space quantization
L has 2l 1 orientations
Electron spin Pauli operator
Stern-Gerra experiment
Spin magnetic moment
Magnetic moment related to electron spin angular momentum
electron spin hypothesis
spin operator
The spin angular momentum operator can only take
is also its eigenvalue
Spin angular momentum square operator
s=1/2 is the spin (angular momentum) quantum number
Pauli matrix
Quantum Measurement Conservation Laws in Quantum Mechanics
The fifth postulate of quantum mechanics--Quantum measurement postulate
The process of measurement is the preparation process of new state
statistical causation
By measuring a system, what we get is not the properties of the original system, but the properties of the system under the action of the measuring instrument.
The material world described by quantum mechanics has only statistical causal connections.
average value of mechanical quantities
The average value of mechanical quantities changes with time. The conservation laws and conserved quantities of quantum mechanics
The average value of the total derivative operator in any state is equal to the derivative of the average value with respect to time
A necessary and sufficient condition for a mechanical quantity to be a conserved quantity of a certain system is that the mechanical operator does not explicitly contain time and is commutative with the Hamiltonian of the system.
The Hamiltonian obviously satisfies the above conditions, so it is the law of conservation of energy in quantum mechanics.
Atomic structure
Central force field problem in quantum mechanics
Quantum state of particles moving in central force field
Solution of the stationary Schrödinger equation in central force field
This equation can only be satisfied by discrete energies
The energy dispersion is represented by nr and is called the radial quantum number.
probability density of particle position
Hydrogen atoms and hydrogen-like ions
Hamiltonian and electronic state of hydrogen atom
Solution to the stationary Schrödinger equation of hydrogen atom
Hydrogen atom energy level structure and spectrum
Probability density of radial position of hydrogen atom electrons
Pauli principle spin wave function of two electrons
The indistinguishability of microscopic particles
Difficulty distinguishing particles in overlapping regions, called inseparable deformation
Symmetric wave function
The spin quantum number of photons and hydrogen atoms in the ground state is zero or a positive integer.
boson
antisymmetric wave function
The spins of electrons, protons, neutrons, etc. are half integers
Fermions
The sixth basic postulate of quantum mechanics
The wave function that describes the state of a system of identical particles is symmetrical to the coordinate exchange of any two particles; the boson system is symmetrical under this exchange, and the fermion wave function is antisymmetric under this exchange.
Pauli's principle of wave function for identical particle systems
No two particles in an identical fermion system can be in the same quantum state
Pauli Exclusion Principle
Bosonic systems can be in the same quantum state
Liquid nitrogen atoms are all in the ground state and exhibit superfluidity, which is called a Bose-Einstein condensate.
Wave function of a two-electron system
Atomic shell structure
Central force field approximation Independent electron model
Atomic shell structure
In ground state atoms, electrons occupy the state with the lowest total energy without violating the constraints of Pauli's principle.
For the same cosmic electron, it can be divided into different branch shells according to different angular quantum values.
A specific shell is called an electron orbital
A shell whose electron occupancy reaches the maximum number of electrons it can accommodate is a closed shell, otherwise it is an open shell.
Quantum mechanical explanation of the periodic law of elements
The periodicity shown by the elements arranged according to the description is actually the result of the periodic distribution of electrons in the atoms in the shell.