量子力学基础

The three-dimensional version of Schrödinger equation:
The total wavefunction is a simple product:
If ψ(x,y,z) = 0 at x=a, ⇒ kxa = n1π, with n = 1,2,3... Similarly, if ψ(x,y,z) = 0 at y = b and z = c: and and
Plane wave(平面波)
For the light wave, the electric field Ey at position x at time t is described by:
A more generalized form is used to describe a plane wave in x direction.
Since V = 0:
Define k2= Solving the differential equation:
Multiplying exp(-jEt/ћ) and ω=E/ћ:
The probability distribution of the electron:
Example 2. Electron in a one-dimensional
Probability interpretation (几率解释）
At any time the particle must certainly be somewhere. The probability of finding the particle with x coordinate between minus and plus infinity must be unity （1）. Hence the wavefunction must have its square modulus integrable and be normalized.（模的平方可积，归一化）
⇒
For a square box for which a=b=c, the energy is
Where N2 =n12 +n22 +n32
The energy is dependent on three quantum numbers. There are three quantum numbers, each one arising from boundary condition along one of the coordinates. The lowest energy for the electron is equal to E111, not zero. The next energy level corresponds to E211, which is the same as E121 and E112, so there are three states (i.e., ψ211, ψ121, ψ112) for the energy. The number of states that have the same energy is termed the degeneracy of the energy level. The second energy level E211 is thus threefold degenerate.
Example 3. Electron confined in three dimensions by
a three dimensional infinite “PE box“(三维无线深势阱)
V=0 in 0 < x < a, 0<y<b and 0< z < c V = ∞, outside
Everywhere inside the box, V = 0, but outside, V = ∞. The electron cannot escape from the box. What is the energy and wavefunction of the electron?
Electron in a one-dimensional infinite PE well. The energy of the electron is quantized. Possible wavefunctions and the probability distributions for the electron are shown.
The Schrödinger equation in the region 0<x<a:
The general solution is: ⇒ From ψ(0)=0
⇒
Note: ejθ = cosθ + j sinθ with j2 = -1
Substitute
in
⇒
⇒
Since no PE (potential energy = 0)
n=1,2,3...
From kn = nπ/a, eigenenergies are:
n=1,2,3...
Normalization condition: The total probability of finding the electron in the whole region 0 < x < a is unity (1).
量子力学基础 De Broglie relationship(德布罗意关系)
(普朗克常量)
Wave-particle duality(波粒二象性) (波矢)
(角频率)
Question (energy of blue photon): what is the energy of a blue photon that has a wavelength of 450 nm?
The wave equation of electrons
The wave equation of electrons
（与时间有关的薛定谔方程）
哈密顿函数
The wavefunction is a solution of the timedependent Schrodinger equation, which determines the wavefunction evolution in space and time
infinite PE well（一维无限深势阱）
Consider the behavior of the electron when it is confined to a certain region, 0 < x < a. Its PE is zero inside that region and infinite outside. The electron cannot escape.
Question (X-ray energy and momentum): X-rays are photons with very short wavelengths that can penetrate or pass through objects, which is used in medical imaging, security scans at airport, x-ray diffraction studies of crystal structures. Typical X-rays have a wavelength of about 0.6 angstrom (1 Å = 10-10 m). Calculate the energy and momentum of an X-ray with this wavelength.
The solution is sin(ka) = 0
⇒
ka=nπ, where n=0,1,2,3,... an Integer (but n=0 is excluded)
K and E are quantized. n is called a quantum number. For each n, there is a special wavefunction (called eigenfunction)
Where n1, n2 and n3 are quantum numbers.
The eigenfunctions of electron, denoted by the quantum numbers n1, n2 and n3, are given by:
Each possible eigenfunction can be labeled a state for the electron. Thus, ψ111 and ψ121 are two possible states. Normalization of |ψn1n2n3(x,y,z)|2 results in A = (2/a)3/2 for a square box (a=b=c). The energy as a function of kx, ky and kz:
Probability interpretation (几率解释）
A particle (e.g. an electron) is described by a complex wavefunction
Ψ(x,t)
The wavefunction must be a continuous, single-valued function of position and time. （波函数单值、连续）
Question (electron confined within atomic dimensions): Consider an electron in an infinite potential well of 0.1 nm (typical size of an atom). What is the ground energy of the electron? What is the energy required to put the electron at the third energy level? How can this energy be provided?