Shulou(Shulou.com)11/24 Report--

By the early 20th century, surprising new experimental evidence began to emerge, challenging the widely accepted view that light consists only of electromagnetic waves (governed by the Maxwell equation). Max Planck's theory of blackbody radiation, which provided a solution to the UV disaster and won him the Nobel Prize in 1918, seems to imply the discreteness and granularity of light. Einstein's theoretical explanation of the photoelectric effect further supported this and won the Nobel Prize in 1921. In their theory, Planck and Einstein proposed that although light has wavy characteristics, it is emitted, propagated and absorbed in the form of discrete energy "packets" or "quantum", which is later called photons. Its energy is given by the Planck-Einstein relationship:

Where 𝜔 represents the angular frequency of the electromagnetic wave, E represents the energy of the corresponding photons, and ℏis only a proportional constant called Planck's constant.

The particle-like behavior of light was decisively proved by Arthur Compton in 1923. To confirm the particle theory of light proposed by Einstein and Planck, Compton conducted a series of experiments in which he scattered x-rays from free electrons. He observed that when the x-rays were scattered, the frequency seemed to suddenly drop. This phenomenon is now called the Compton effect.

Compton found that his observations could not be explained by the classical description of light and electromagnetic waves, because the frequency of the classical light and electromagnetic waves did not change when they predicted scattering. However, if he adopts the photon theory of Einstein and Planck and believes that an x-ray is an inelastic collision between a particle and a free electron and loses energy in the process, this effect can be perfectly explained. The resulting conflict between light waves and particle descriptions is called wave-particle duality theory.

In the 1920s, however, more startling evidence appeared, which seemed to prove not only the particle-like behavior of light, but also the wavy behavior of matter particles. This began with a variety of electron diffraction experiments, such as the 1927 Davidson-Gormer experiment, in which electrons were scattered from the crystal surface, showing a diffraction pattern. The diffraction pattern can only be produced by the interference effect between the propagating waves. Therefore, the results of the Davidson-Gomer experiment can only be explained by the mathematical method of propagating waves to describe the scattered electrons.

Animation of electron diffraction patterns in order to explain more and more evidence to support the wavy properties of electrons, physicist Louis de Broglie proposed in his doctoral thesis on Quantum Theory in 1924 that every particle of matter has a related wave. this wave was later called a matter wave or de Broglie wave with a corresponding de Broglie wavelength. This matter wave will then capture the wavy aspect of the particle's behavior. De Broglie also found that this concept of matter waves can also provide an explanation for the observed atomic discrete spectrum, indicating that there are only discrete, allowable electron orbitals.

The electron can no longer be thought of as a single small particle; it must be related to a wave, which is not a myth; its wavelength can be measured and its interference can be predicted-Louis de Broglie, 1929 Nobel Prize speech

In order to bridge the gap between the description of particles and waves of matter, de Broglie designed the relationship between the properties of particles and the properties of their corresponding matter waves. He did this by associating the momentum of particles with the de Broglie wavelength of their matter waves.

Where p represents the momentum of the particle, 𝝀 is the de Broglie wavelength of the corresponding matter wave, and k is the "wave vector" or "wave number" of the matter wave. In the three spatial coordinates, this is usually written as:

Where bold symbols represent vectors. This relationship is called the de Broglie relationship.

De Broglie relation and Einstein-Planck relation build a bridge between the wave of matter and the description of particles by linking the energy and momentum of particles with the frequency and wavelength of their related waves. These two relationships form the basis of early quantum theory.

Now let's consider a simple one-dimensional de Broglie "plane wave" defined by amplitude A, wave vector k, and angular frequency 𝜔. This wave, which we call a wave function from here, can be expressed as:

We use Euler identity to represent the complex index in trigonometric form. For the sake of simplicity, we only consider the propagation of plane waves in the positive x direction. To illustrate this point, we can draw the real and virtual components of matter waves at the initial time t:

However, the actual quantity represented by 𝚿 (x, t) can only be interpreted as probabilistic amplitude. This interpretation is called born probabilistic amplitude interpretation.

We can now use de Broglie and Einstein-Planck relations to associate the properties of the wave function with the properties of the corresponding particles. Get:

Now that we have a description of the matter wave of the particle, let's also define the total energy of the particle. In general, the total energy consists of kinetic and potential energy terms, as follows:

Where m is the mass of the particle, p is the momentum of the particle, and V (x) represents the potential field related to a certain space.

Schrodinger's paper in 1926, he first published the time-dependent Schrodinger equation. Schrodinger published the Schrodinger equation in his 1926 paper "Wave Theory of Atomic and Molecular Mechanics", which is a differential equation. it describes how the wave function that represents the state of particles evolves over time. Therefore, let's calculate the time derivative of the matter wave or wave function 𝚿 (x, t) expression and get the following results:

If we put the total energy of the particles into the expression, we get:

Expression 1 now, all that's left is to express momentum p in terms of position x. In order to do this, we calculate the second-order spatial derivative of the wave function 𝚿 (x, t).

If we divide by 2m on both sides, and then rearrange it, we get:

Or in the operator representation, we pull out the wave function term:

Therefore, we can substitute this expression into the time derivative in expression 1 to get:

This just gives the one-dimensional Schrodinger equation which varies with time. More generally, in three-dimensional space, we can simply replace x with the position vector r and the partial derivative of x with the Laplace function.

This will provide us with the following representation of the Schrodinger equation over time:

The time-dependent Schrodinger equation is the most general form of the Schrodinger equation, from which all other forms can be derived. By considering the time-independent potential field and the time-independent Hamiltonian, the time-independent form of the Schrodinger equation can be derived.

De Broglie's hypothesis is an extremely important stepping stone to some of the most basic concepts in quantum mechanics. In addition to the Schrodinger equation, we can also use the concept of matter wave to derive Heisenberg's quantum uncertainty principle directly.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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