# 物理代写|量子力学代写quantum mechanics代考|Massive Scalar Field

## 物理代写|量子力学代写quantum mechanics代考|Massive Scalar Field

The relativistic relation between energy and momentum for a particle with rest mass $m_0$ is
$$E=\sqrt{\vec{p}^2 c^2+m_0^2 c^4}$$
If we identify this with the hamiltonian $H$, and attempt to quantize with $\vec{p}=(\hbar / i) \vec{\nabla}$, the Schrödinger equation becomes
$$i \hbar \frac{\partial \phi(\vec{x}, t)}{\partial t}=\left[-(\hbar c)^2 \vec{\nabla}^2+\left(m_0 c^2\right)^2\right]^{1 / 2} \phi(\vec{x}, t)$$

The square-root causes difficulties. ${ }^1$ A repeated application of this expression, however, leads to a much simpler expression
$$\left(\square-m^2\right) \phi(\vec{x}, t)=0 \quad ; m \equiv \frac{m_0 c}{\hbar}$$
This is the relativistic wave equation for a particle with inverse Compton wavelength $m=m_0 c / \hbar$.

We know from the previous text Introduction to Classical Mechanics [Walecka (2020)] how to do classical continuum mechanics with the wave equation. ${ }^2$ There the analysis is applied to the string, where the wave equation holds. Introduce the two-vector $x_\mu=\left(x_1, x_2\right)=(x, i c t)$, with $c^2=\tau / \sigma$ and $i$ the imaginary number $\sqrt{-1}$. Also, introduce the convention that repeated Greek indices are summed from 1 to 2 . The basic equation of motion is then obtained from Hamilton’s principle of stationary action
$$\delta \int d^2 x \mathcal{L}\left(q, \frac{\partial q}{\partial x_\mu}\right)=0 \quad ; d^2 x \equiv d x c d t$$
The lagrangian density for the string is
$$\mathcal{L}=\frac{\sigma}{2}\left[\frac{\partial q(x, t)}{\partial t}\right]^2-\frac{\tau}{2}\left[\frac{\partial q(x, t)}{\partial x}\right]^2$$

## 物理代写|量子力学代写quantum mechanics代考|The Dirac Equation

The first successful union of quantum mechanics and special relativity for a single-particle was achieved by Dirac, and here we give the lovely historical argument [Dirac (1926)].
One wants the theory to possess the following features:
(1) A positive-definite probability density
$$\rho=\Psi^{\star} \Psi \geq 0 \quad \text {; probability density }$$

(2) A Schrödinger equation that is first-order in the time derivative
$$i \hbar \frac{\partial \Psi}{\partial t}=H \Psi \quad \text {; Schrödinger equation }$$
(3) A continuity equation
$$\frac{\partial \rho}{\partial t}+\vec{\nabla} \cdot \vec{S}=0 \quad \text {; continuity equation }$$
As before, this will provide a basis for the interpretation of the theory and ensure that, for a localized particle,
$$\frac{d}{d t} \int \rho(\vec{x}, t) d^3 x=0$$
(4) The correct relativistic relation between energy and momentum
$$E^2=\vec{p}^2 c^2+m_0^2 c^4 \quad ; \text { relativistic relation }$$
Now we do know of a theory that is Lorentz covariant and involves only first-order time derivatives, and that is the set of Maxwell’s equations in electrodynamics.8 Here one has a set of eight coupled equations for the components of the electric and magnetic fields $(\vec{E}, \vec{B})$. Dirac argued by analogy. He introduced a wave function $\Psi$ that had a set of $n$ components
$$\psi_\sigma \quad ; \sigma=1,2, \cdots, n$$
components of $\Psi$
with a corresponding positive-definite probability density defined by
$$\rho \equiv \sum_{\sigma=1}^n \psi_\sigma^{\star} \psi_\sigma$$

# 量子力学代考

## 物理代写|量子力学代写quantum mechanics代考|Massive Scalar Field

$$E=\sqrt{\vec{p}^2 c^2+m_0^2 c^4}$$

$$i \hbar \frac{\partial \phi(\vec{x}, t)}{\partial t}=\left[-(\hbar c)^2 \vec{\nabla}^2+\left(m_0 c^2\right)^2\right]^{1 / 2} \phi(\vec{x}, t)$$

$$\left(\square-m^2\right) \phi(\vec{x}, t)=0 \quad ; m \equiv \frac{m_0 c}{\hbar}$$

$$\delta \int d^2 x \mathcal{L}\left(q, \frac{\partial q}{\partial x_\mu}\right)=0 \quad ; d^2 x \equiv d x c d t$$

$$\mathcal{L}=\frac{\sigma}{2}\left[\frac{\partial q(x, t)}{\partial t}\right]^2-\frac{\tau}{2}\left[\frac{\partial q(x, t)}{\partial x}\right]^2$$

## 物理代写|量子力学代写quantum mechanics代考|The Dirac Equation

（1）正定概率密度
$$\rho=\Psi^{\star} \Psi \geq 0 \quad ; \text { probability density }$$
(2) 时间㝵数为一阶的薛定遌方程
$i \hbar \frac{\partial \Psi}{\partial t}=H \Psi \quad ;$ Schrödinger equation
(3)一个连续性方程
$$\frac{\partial \rho}{\partial t}+\vec{\nabla} \cdot \vec{S}=0 \quad ; \text { continuity equation }$$

$$\frac{d}{d t} \int \rho(\vec{x}, t) d^3 x=0$$
(4) 能量与动量的正确相对论关系
$$E^2=\vec{p}^2 c^2+m_0^2 c^4 \quad ; \text { relativistic relation }$$

$$\psi_\sigma \quad ; \sigma=1,2, \cdots, n$$

$$\rho \equiv \sum_{\sigma=1}^n \psi_\sigma^{\star} \psi_\sigma$$

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