⑤ Systems for which exact solutions of Schrödinger equation can be obtained : Simple harmonic oscillation, particle in a 1-dimensional box, rigid body rotation, hydrogen atom, etc.

⑥ Systems for which exact solutions of Schrödinger equation cannot be obtained : Helium atom, etc.

2. Uncertainty Principle

⑴ Presented by Heisenberg (Werner Karl Heisenberg)

⑵ Proof: Using Schrödinger Equation and Cauchy-Schwarz inequality

Figure. 1. Proof of Uncertainty Principle [Note:1]

⑶ Simultaneously knowing momentum and position accurately is impossible

① Formulation

② (Note) Momentum is derived from symmetry with respect to space and is related to space

③ Application: Uncertainty Principle and Microscopy

Figure. 2. Uncertainty Principle and Microscopy [Note:2]

○ Momentum of a photon with wavelength λ is h/λ, so decreasing λ increases photon’s momentum

○ Larger momentum leads to larger Δp of the electron

○ Conclusion: Decreasing λ to improve microscope resolution increases Δp, making it harder to know momentum

⑷ Simultaneously knowing energy and time accurately is impossible

① Formulation

② (Note) Energy is derived from symmetry with respect to time and is related to time

⑸ Phase Velocity

① Quantum mechanical concept of confined classical particle in a given region

② Linear combination of solutions of wave functions of particles with certain energy within that region

③ Group Velocity: Concept of speed when multiple wave functions overlap

④ Dispersion Relationship

3. Free Particle : U = 0

⑴ Conclusion 1: ψ is derived from wave equation

⑵ Conclusion 2: Momentum follows de Broglie’s relation

⑶ Conclusion 3: Energy is same as that of a free particle in classical mechanics

4. Particle in a Box (generally particle in one dimensional box)

⑴ Assumptions

① x < 0, x > L : Potential energy is ∞

② 0 ≤ x ≤ L : Potential energy is 0

③ Particle is considered as a matter wave, satisfying normal wave conditions, i.e., nλ = 2L

④ Wave function is independent of time

⑵ Key Formulas

① Probability amplitude function

② Energy

⑶ Example 1: In the wave function of an electron confined in a 1-dimensional infinite potential well with width L, derived from normalization of probability, let the amplitude be denoted as A. If the well’s width is halved, what is the amplitude of the wave function?

⑷ Example 2: If ψn(x) is the normalized nth eigenstate wave function, what is the energy of a particle with the following state function?

① Idea: Given particle has a probability of 2/3 for the 1st eigenstate and 1/3 for the 2nd eigenstate

② Answer: E = (2/3) × E2 + (1/3) × E2 = 2E2

5. Tunneling Effect

⑴ Definition : Phenomenon where a particle passes through a barrier it couldn’t pass if treated as a particle

⑵ Solving the Schrödinger equation shows that there’s a probability of finding the particle outside a finite potential well

⑶ Tunneling Effect Problem

① Question: A particle with energy 0.1 eV is incident on a potential barrier with a thickness of 2 nm and a height of 4 eV. The probability of the particle tunneling through the barrier is T0. Under the same conditions, if the thickness of the barrier is changed to 3 nm, what is the probability of the particle tunneling through?

Figure. 3. Tunneling Effect Problem [Note:3]

② Answer: The probability of the particle tunneling through 6 nm is the cube of the probability of tunneling through 2 nm and the square of the probability of tunneling through 3 nm

⑷ Application : Scanning Tunneling Microscope (STM)

Figure. 4. Scanning Tunneling Microscope [Note:4]

① The potential barrier is wider when the probe is at P compared to when it’s at Q

② The probability of the electron moving between the probe and the sample is lower when the probe is at P compared to when it’s at Q ( ∵ due to the wider barrier)

③ The current is weaker when the probe is at P compared to when it’s at Q (∵ due to the lower probability of electron movement)

④ The surface structure of the sample can be determined by reading the current

6. Effective Mass

7. Reflection of Matter Waves

⑴ Reflection of Electromagnetic Waves : Reflection Coefficient = Reflected Intensity / Incident Intensity (where η = Intrinsic Impedance)

⑵ Reflection of Matter Waves

Figure. 5. Situation of Reflection of Matter Waves Problem [Note:5]

① Situation 1: Particle beam with E > 0 is incident from the left

② Situation 2: Momentum of particles is conserved as ℏk in each region

③ Reflection coefficient R at x = 0 is as follows

8. Coordinate System Transformation

⑴ r : Distance from the origin

⑵ θ : Angle from the z-axis (latitude)

⑶ φ : Angle around the z-axis (longitude)

⑷ R(r) (radial wave function) : Determines orbital size

⑸ Y(θ, φ) (angular wave function) : Determines orbital direction and distribution shape

9. Solutions of Wave Equations

⑴ Wave function is derived only when quantized variables (= quantum numbers) are determined

⑵ Solutions of radial wave function

⑶ Solutions of angular wave function

Input: 2019-09-08 21:04

Modification: 2020-02-08 23:45

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Chapter 5. Quantum Mechanics Part 2

1. Schrödinger Equation