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The charm of the time-independent Schrödinger equation: Do you know how it explains the behavior of particles?
In the field of quantum mechanics, the Time-Independent Schrödinger Equation (TISE) is a basic tool used to describe the behavior of particles in a specific potential field. Among them, the one-dimensional step potential energy problem is an idealized system used to simulate the incident, reflected and transmitted matter waves. This article will explore in depth how this equation helps us understand the behavior of particles in step potential and reveal the quantum mysteries involved.
Schrödinger equation and potential function
The time-independent Schrödinger equation can be expressed as:
H ^ ψ(x) = [ - ℏ² / (2m) d² / dx² + V(x) ] ψ(x) = E ψ(x)
Here, H is the Hamiltonian, ℏ is the reduced Planck constant, m is the mass of the particle, and E is the energy of the particle. For one-dimensional step potential energy, the potential function is usually expressed as a Heaviside step function:
V(x) = { 0 , x < 0; V0 , x ≥ 0 }
This means that when x is less than 0, the particle has no potential, and when x is greater than or equal to 0, the particle moves under the influence of the potential V0. Such a setup allows us to analyze the behavior of particles in different regions and lays the foundation for our research.
Solution
In a step potential, space is divided into two regions: x < 0 and x > 0. In both regions, the potential energy is constant, which means that the particles are quasi-free in these regions. Here, the solutions to the Schrödinger equation can be expressed as the superpositions of the left and right moving waves, which can be written as:
ψ₁(x) = (A→ e^(ik₁x) + A← e^(-ik₁x)) x < 0
ψ₂(x) = (B→ e^(ik₂x) + B← e^(-ik₂x)) x > 0
Here, A and B represent the amplitude of the wave, the directional arrows represent the direction of motion, and k₁ and k₂ are the wave numbers corresponding to different energies, respectively.
Boundary Conditions
The coefficients A and B of the wave function need to be determined based on the boundary conditions at x=0. In order to ensure the continuity of the wave function and its derivatives at the boundary, it is necessary to set the following conditions:
ψ₁(0) = ψ₂(0)
dψ₁/dx|_{x=0} = dψ₂/dx|_{x=0}
Such boundary conditions provide explicit constraints on our coefficients, allowing us to calculate the probabilities of reflection (R) and transmission (T).
Transmission and reflection
In quantum mechanics we can see a contrast with the classical situation. A particle can be reflected or teleported when it comes into contact with a step potential. Assuming that the particle energy E is greater than V0, the particle incident from the left side A can be reflected (A←) or transmitted (B→).
R = (k₁ - k₂)/( k₁ + k₂ )
T = 2√(k₁*k₂)/(k₁ + k₂)
These formulas reveal the nature of quantum particles' interaction with potential, especially their behavior when the particle energy is higher than the potential, which makes the calculation of the probability of transmission and reflection particularly interesting.
In-depth analysis
The analysis is not limited to the above case. When the energy is less than the step height (E < V0), the wave function on the right will decay exponentially. This behavior does not appear in classical physics. Furthermore, when the energy is greater than the step height, the results of transmission and reflection are contrary to classical insights, which has led to the exploration of phenomena such as the Klein Paradox.
Application and Extension
The step potential model is mainly used in introductory quantum mechanics textbooks to help students understand several important concepts such as the regularization of wave functions, boundary conditions, entry/reflection/transmission amplitudes and their probabilities. Furthermore, variants of this problem also find a place in superconducting metal interface physics, where quasiparticles scatter on pairing potentials with a step shape, which has mathematical similarities to the step potential problem in question.
With the development of quantum mechanics, the time-independent Schrödinger equation remains one of the important tools for exploring the microscopic world. As our understanding of quantum phenomena deepens, are you also wondering how these phenomena affect the laws of physics in our daily lives?
