“Physics Memo” has been used by many people since it was opened in November 2017, but due to various reasons, it will be closed on February 28, 2021.
We would like to express our deepest gratitude for your patronage.
How to Derive the Schrödinger Equation and Momentum Operator
The Schrödinger Equation
$$iħ\frac{∂}{∂t}Ψ({\bf r},t)=-\frac{ħ^2}{2m}∇^2Ψ({\bf r},t)$$
Using Hamiltonian,
$$iħ\frac{∂}{∂t}Ψ({\bf r},t)=\hat{H}Ψ({\bf r},t)$$
In this post, after deriving the Schrödinger Equation equation, we obtain the momentum operator.
Lorentz Contraction and its Derivation
Observe moving objects near the light speed from a static system. Then, the object looks shorter than the original length (the length when the object stopped). This phenomenon is called Lorentz contraction.
Let \(L_0\) be the length of the bar when the object is stopped and \(L\) the length of the moving bar observed from the static system. Then, the following relationship holds between the two. However, it is assumed that the system S’ is moving at a speed V in the x direction with respect to the system S and the bar is moving with the system S’.
\begin{eqnarray} L&=&L_0\sqrt{1-\frac{V^2}{c^2}}\\&=&L_0\sqrt{1-β^2} \end{eqnarray}
Proofs of Kepler’s Laws
Kepler’s law is what concerning the movement of a planet. If you can master this rule, you can easily think about the movement of the planet. And in order to think about this, it is the quickest to introduce the motion equation of two-dimensional polar coordinates. In this post, I’ll introduce Kepler’s law and prove it.
Universal Gravitation and Gravitational Potential – How to find escape speed
The universal gravitational force working between a particle of mass \(m\) and a particle of mass \(M\) is expressed as follows when the distance between the particles is \(r\).
$$f(r)=-G\frac{mM}{r^2}$$
$$G=6.672 \times 10^{-11}[N･m^2/kg^2]$$
\(G\) is called gravitational constant.
Meaning of Bra-ket Notation
In quantum mechanics, use bra vector \(<a|\) and ket vector \(|a>\). What are the advantages of introducing them?
Exercise of de Broglie wave
In quantum mechanics, we believe that every object has both particle and wave properties. If so, there should be rules for linking the properties of the particles and those of the waves.
De Broglie suggested the following equation, thinking that particles with masses such as protons and electrons can also be regarded as waves.
$$λ=\frac{h}{p}$$
\(h\) is Planck’s constant, \(p\) is the particle momentum. Then the particle has the same properties as wave of wavelength \(λ\). This wavelength \(λ\) is called the de Broglie wavelength.
Why Do We Need to Solve Schrödinger’s Equation?
[mathjax]
In this post, we’ll discuss what solving Schrödinger’s equation brings us by considering the physical meaning of the wave function.
Method of Setting Integral Path of Complex Integral
[mathjax]
I described a method of solving complex integrals by theorems before, but some complex integrals can not be solved only applying theorems. However, even with such complex integrals, it becomes possible to solve by setting an integration route yourself.
Furthermore, real integrals can be solved as complex integrals. In this article, I’ll write about some integrals that can be solved by applying this.
Lorentz Transformation
[mathjax]
Most relativity theory book starts with Lorentz transformation. Lorentz transformation is to connect time and space of two systems that have different motion.
Unlike the classical mechanics and electromagnetism we have learned so far, the theory of relativity is incredible at first sight. For example, according to the theory of relativity, an object with mass can not catch up with the speed of light. In addition, if you are moving at a speed close to light, the time progresses slowly for you. Let’s consider the meaning of Lorentz transformation carefully by reading this article.