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Physics Appendix 📂Mathematical Physics

Physics Appendix

Fundamentals

Appendix A-1

d∣x∣dx=dx2dx=dx2dx2dx2dx=121x2⋅2x=1∣x∣x \begin{align*} \frac{ d |x| } {d x} &= \frac{d \sqrt{x^2} }{d x} \\ &= \frac{d \sqrt{x^2}}{d x^2} \frac{d x^2}{dx} \\ &= \frac{1}{2}\frac{1}{\sqrt{x^2}} \cdot 2x \\ &= \dfrac{1}{|x|}x \end{align*}

Electromagnetism

Appendix E-1

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δ(f(x))=∑x0δ(x−x0)∂f∂x∣x=x0 \delta \big( f(x) \big) =\sum \limits_{x_{0}} \frac{\delta (x-x_{0})}{ \frac{\partial f}{\partial x}\Big|_{x=x_{0}} }

In this case, x0x_{0} is a solution to f(x)f(x). Using the fact above,

δ(t′−t+∣r−w(t′)∣c)=δ(t′−tr)1+∂∂t′(∣r−w(t′)∣c)∣t′=tr \delta \left( t^{\prime}-t+\frac{ | \mathbf{r} -\mathbf{w}(t^{\prime}) | }{c} \right) = \frac{ \delta (t^{\prime}-t_{r}) }{ 1+\frac{\partial}{\partial t^{\prime}}\left( \frac{|\mathbf{r}-\mathbf{w}(t^{\prime})|}{c} \right)\Big| _{t^{\prime}=t_{r}} }

Since ddxf(g(x))12=12f(g(x))g′(x)\frac{ d}{dx} f(g(x))^{\frac{1}{2}} =\dfrac{1}{2\sqrt{ f(g(x)) }}g^{\prime}(x), if we unravel the denominator on the right side,

1+∂∂t′(∣r−w(t′)∣c)=1+1c∂∂t′([(r−w(t′))⋅(r−w(t′))]12)=1+1c12(r−w(t′))⋅(r−w(t′))∂∂t′((r−w(t′))⋅(r−w(t′)))=1+1c12∣r−w(t′)∣2(r−w(t′))⋅∂∂t′(r−w(t′))=1+r−w(t′)c∣r−w(t′)∣⋅(−v(t′))=1−[r−w(t′)]⋅v(t′)c∣r−w(t′)∣ \begin{align*} 1+\frac{\partial}{\partial t^{\prime}}\left( \frac{|\mathbf{r}-\mathbf{w}(t^{\prime})|}{c} \right) &= 1+\frac{1}{c} \dfrac{ \partial }{\partial t^{\prime}} \left( \left[\big( \mathbf{r}-\mathbf{w}(t^{\prime}) \big) \cdot \big( \mathbf{r}-\mathbf{w}(t^{\prime}) \big) \right]^{\frac{1}{2}} \right) \\ &= 1+\frac{1}{c} \dfrac{1}{2 \sqrt{ \big( \mathbf{r}-\mathbf{w}(t^{\prime}) \big) \cdot \big( \mathbf{r}-\mathbf{w}(t^{\prime}) \big)} } \dfrac{\partial}{\partial t^{\prime}} \bigg( \big( \mathbf{r}-\mathbf{w}(t^{\prime}) \big)\cdot \big( \mathbf{r}-\mathbf{w}(t^{\prime}) \big) \bigg) \\ &= 1+\frac{1}{c} \dfrac{1}{2 |\mathbf{r}-\mathbf{w}(t^{\prime})|} 2\big(\mathbf{r}-\mathbf{w}(t^{\prime}) \big)\cdot\dfrac{\partial}{\partial t^{\prime}} \big( \mathbf{r}-\mathbf{w}(t^{\prime}) \big) \\ &= 1 +\frac{ \mathbf{r}-\mathbf{w}(t^{\prime}) }{c | \mathbf{r}-\mathbf{w}(t^{\prime})|} \cdot \big(-\mathbf{v}(t^{\prime}) \big) \\ &= 1-\frac{ [ \mathbf{r}-\mathbf{w}(t^{\prime}) ]\cdot \mathbf{v}(t^{\prime}) }{c | \mathbf{r}-\mathbf{w}(t^{\prime})|} \end{align*}

and then substitute t′=trt^{\prime}=t_{r},

1−[r−w(tr)]⋅v(tr)c∣r−w(tr)∣=1−η⋅v(tr)cη=1−η^⋅v(tr)c 1-\frac{ [\mathbf{r}-\mathbf{w}(t_{r})] \cdot \mathbf{v}(t_{r}) }{c | \mathbf{r}-\mathbf{w}(t_{r})|} =1-\frac{ \boldsymbol{\eta} \cdot \mathbf{v}(t_{r}) }{c \eta} = 1-\frac{ \hat{\boldsymbol{\eta}} \cdot \mathbf{v}(t_{r}) }{c }

Therefore,

δ(t′−t+∣r−w(t′)∣c)=δ(t′−tr)1−η^⋅v(tr)c \delta \left( t^{\prime}-t+\frac{ | \mathbf{r} -\mathbf{w}(t^{\prime}) | }{c} \right) =\frac{ \delta (t^{\prime}-t_{r}) } {1-\frac{ \hat{\boldsymbol{\eta}} \cdot \mathbf{v}(t_{r}) }{c } }

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Quantum Mechanics

Appendix Q-1

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A=(e−ikaeikaike−ika−ikeika),B=(eκae−κaκeκa−κe−κa) \mathbb{A} = \begin{pmatrix} e^{-ika} & e^{ika} \\ ike^{-ika} & -ike^{ika} \end{pmatrix},\quad \mathbb{B} = \begin{pmatrix} e^{\kappa a} & e^{-\kappa a} \\ \kappa e^{\kappa a} & -\kappa e^{-\kappa a} \end{pmatrix}

By calculating the inverse matrices of the two matrices using minors,

A−1=1∣A∣adj(A)=1−ik−ik(−ikeika−ike−ika−eikae−ika)T=1−2ik(−ikeika−eika−ike−ikae−ika)=12(eika1ikeikae−ika−1ike−ika) \begin{align*} \mathbb{A}^{-1} &= \frac{1}{|\mathbb{A}|} \mathrm{adj}(A) \\ &= \frac{1}{-ik -ik} \begin{pmatrix} -ike^{ika} & -ike^{-ika} \\ -e^{ika} & e^{-ika} \end{pmatrix}^T \\ &= \frac{1}{-2ik} \begin{pmatrix} -ike^{ika} & -e^{ika} \\ -ike^{-ika} & e^{-ika} \end{pmatrix} \\ &= \frac{1}{2} \begin{pmatrix} e^{ika} & \frac{1}{ik}e^{ika} \\ e^{-ika} & \frac{-1}{ik} e^{-ika} \end{pmatrix} \end{align*}

B−1=1∣B∣adj(B)=1−κ−κ(−κe−κa−κeκa−e−κaeκa)T=1−2κ(−κe−κa−e−κa−κeκaeκa)=12(e−κa1κe−κaeκa−1κeκa) \begin{align*} \mathbb{B}^{-1} &= \frac{1}{|\mathbb{B}|} \mathrm{adj}(B) \\ &= \frac{1}{-\kappa -\kappa } \begin{pmatrix} -\kappa e^{-\kappa a} & -\kappa e^{\kappa a} \\ -e^{-\kappa a} & e^{\kappa a} \end{pmatrix}^T \\ &= \frac{1}{-2\kappa } \begin{pmatrix} -\kappa e^{-\kappa a} & -e^{-\kappa a} \\ -\kappa e^{\kappa a} & e^{\kappa a} \end{pmatrix} \\ &= \frac{1}{2} \begin{pmatrix} e^{-\kappa a} & \frac{1}{\kappa }e^{-\kappa a} \\ e^{\kappa a} & \frac{-1}{\kappa } e^{\kappa a} \end{pmatrix} \end{align*}

Then,

(A+A−)=A−1(e−κaeκaκe−κa−κeκa)B−1(eikae−ikaikeika−ike−ika)(C+0)=12(eika1ikeikae−ika−1ike−ika)(e−κaeκaκe−κa−κeκa)12(e−κa1κe−κaeκa−1κeκa)(eikae−ikaikeika−ike−ika)(C+0)=122(eika1ikeikae−ika−1ike−ika)(e−κaeκaκe−κa−κeκa)(e−κa1κe−κaeκa−1κeκa)(C+eikaC+ikeika) \begin{align*} \begin{pmatrix} A_{+} \\ A_{-} \end{pmatrix} &= \mathbb{A}^{-1} \begin{pmatrix} e^{-\kappa a} & e^{\kappa a} \\ \kappa e^{-\kappa a} & -\kappa e^{\kappa a} \end{pmatrix} \mathbb{B}^{-1} \begin{pmatrix} e^{ik a} & e^{-ik a} \\ ik e^{ik a} & -ik e^{-ika } \end{pmatrix} \begin{pmatrix} C_{+} \\ 0 \end{pmatrix} \\ &= \frac{1}{2} \begin{pmatrix} e^{ika} & \frac{1}{ik}e^{ika} \\ e^{-ika} & \frac{-1}{ik} e^{-ika} \end{pmatrix} \begin{pmatrix} e^{-\kappa a} & e^{\kappa a} \\ \kappa e^{-\kappa a} & -\kappa e^{\kappa a} \end{pmatrix} \frac{1}{2} \begin{pmatrix} e^{-\kappa a} & \frac{1}{\kappa }e^{-\kappa a} \\ e^{\kappa a} & \frac{-1}{\kappa } e^{\kappa a} \end{pmatrix} \begin{pmatrix} e^{ik a} & e^{-ik a} \\ ik e^{ik a} & -ik e^{-ika } \end{pmatrix} \begin{pmatrix} C_{+} \\ 0 \end{pmatrix} \\ &= \frac{1}{2^2} \begin{pmatrix} e^{ika} & \frac{1}{ik}e^{ika} \\ e^{-ika} & \frac{-1}{ik} e^{-ika} \end{pmatrix} \begin{pmatrix} e^{-\kappa a} & e^{\kappa a} \\ \kappa e^{-\kappa a} & -\kappa e^{\kappa a} \end{pmatrix} \begin{pmatrix} e^{-\kappa a} & \frac{1}{\kappa }e^{-\kappa a} \\ e^{\kappa a} & \frac{-1}{\kappa } e^{\kappa a} \end{pmatrix} \begin{pmatrix} C_{+}e^{ika} \\ C_{+}ike^{ika} \end{pmatrix} \end{align*}

Let’s first compute the second and third matrices.

(e−κaeκaκe−κa−κeκa)(e−κa1κe−κaeκa−1κeκa)=(e−2κa+e2κa1κ(e−2κa−eκa)κ(e−2κa−e2κa)e−2κa+e2κa)=(2cosh⁡(2κa)−1κ2sinh⁡(2κa)−2κsinh⁡(2κa)2cosh⁡(2κa)) \begin{align*} \begin{pmatrix} e^{-\kappa a} & e^{\kappa a} \\ \kappa e^{-\kappa a} & -\kappa e^{\kappa a} \end{pmatrix} \begin{pmatrix} e^{-\kappa a} & \frac{1}{\kappa }e^{-\kappa a} \\ e^{\kappa a} & \frac{-1}{\kappa } e^{\kappa a} \end{pmatrix} &= \begin{pmatrix} e^{-2\kappa a} +e^{2\kappa a} & \frac{1}{\kappa} (e^{-2\kappa a } - e^{\kappa a} ) \\ \kappa (e^{-2\kappa a } - e^{2\kappa a } ) & e^{-2\kappa a} + e^{2\kappa a} \end{pmatrix} \\ &= \begin{pmatrix} 2\cosh (2\kappa a) & -\frac{1}{\kappa} 2\sinh (2\kappa a) \\ -2\kappa \sinh(2\kappa a ) & 2\cosh (2\kappa a) \end{pmatrix} \end{align*}

Substituting this back into the original equation and calculating,

(A+A−)=12(eika1ikeikae−ika−1ike−ika)(cosh⁡(2κa)−1κsinh⁡(2κa)−κsinh⁡(2κa)cosh⁡(2κa))(C+eikaC+ikeika)=12(eika(cosh⁡(2κa)−κiksinh⁡(2κa))eika(−1κsinh⁡(2κa)+1ikcosh⁡(2κa))e−ika(cosh⁡(2κa)+κiksinh⁡(2κa))e−ika(−1κsinh⁡(2κa)−1ikcosh⁡(2κa)))(eikaikeika)C+ \begin{align*} \begin{pmatrix} A_{+} \\ A_{-} \end{pmatrix} &= \frac{1}{2} \begin{pmatrix} e^{ika} & \frac{1}{ik}e^{ika} \\ e^{-ika} & \frac{-1}{ik} e^{-ika} \end{pmatrix} \begin{pmatrix} \cosh (2\kappa a) & -\frac{1}{\kappa} \sinh (2\kappa a) \\ -\kappa \sinh(2\kappa a ) & \cosh (2\kappa a) \end{pmatrix} \begin{pmatrix} C_{+}e^{ika} \\ C_{+}ike^{ika} \end{pmatrix} \\ &= \frac{1}{2} \begin{pmatrix} e^{ika} \big(\cosh (2\kappa a) -\frac{\kappa}{ik}\sinh (2\kappa a) \big) & e^{ika} \big( -\frac{1}{\kappa} \sinh (2\kappa a) + \frac{1}{ik}\cosh (2\kappa a) \big) \\ e^{-ika} \big( \cosh (2\kappa a ) + \frac{\kappa}{ik} \sinh (2\kappa a ) \big) & e^{-ika} \big( -\frac{1}{\kappa } \sinh (2\kappa a) - \frac{1}{ik}\cosh (2\kappa a) \big) \end{pmatrix} \begin{pmatrix} e^{ika} \\ ik e^{ika} \end{pmatrix} C_+ \end{align*}

Then,

A+=12[ei2ka(cosh⁡(2κa)−κiksinh⁡(2κa))+ikei2ka(−1κsinh⁡(2κa)+1ikcosh⁡(2κa))]C+=12ei2ka[cosh⁡(2κa)−κiksinh⁡(2κa)−ikκsinh⁡(2κa)+cosh⁡(2κa)]C+=12ei2ka[2cosh⁡(2κa)+i(κk−kκ)sinh⁡(2κa)]=12ei2ka(2cosh⁡(2κa)+i2ηsinh⁡(2κa))C+=ei2ka(cosh⁡(2κa)+iηsinh⁡(2κa))C+ \begin{align*} A_{+} &= \frac{1}{2} \left[ e^{i2ka} \big(\cosh (2\kappa a) -\frac{\kappa}{ik}\sinh (2\kappa a) \big) + ike^{i2ka} \big( -\frac{1}{\kappa} \sinh (2\kappa a) + \frac{1}{ik}\cosh (2\kappa a) \big) \right]C_{+} \\ &= \frac{1}{2} e^{i2ka} \bigg[\cosh (2\kappa a) -\frac{\kappa}{ik}\sinh (2\kappa a) -\frac{ik}{\kappa} \sinh (2\kappa a) + \cosh (2\kappa a) \bigg] C_{+} \\ &= \frac{1}{2} e^{i2ka} \left[ 2\cosh (2\kappa a ) + i\left( \frac{\kappa}{k} -\frac{k}{\kappa} \right) \sinh (2\kappa a) \right] \\ &= \frac{1}{2} e^{i2ka} \Big( 2\cosh (2\kappa a) +i2\eta \sinh (2\kappa a) \Big) C_+ \\ &= e^{i2ka} \Big( \cosh (2\kappa a ) + i\eta \sinh (2\kappa a) \Big) C_+ \end{align*}

This time, it was substituted by κk−kκ=2η\dfrac{\kappa}{k}-\dfrac{k}{\kappa}=2\eta. Therefore,

C+A+=e−i2kacosh⁡(2κa)+iηsinh⁡(2κa) \dfrac{C_{+}} {A_{+}} = \frac{e^{-i2ka}} {\cosh (2\kappa a) + i\eta \sinh (2\kappa a) }

Also,

A−=12[cosh⁡(2κa)+κiksinh⁡(2κa)−ikκsinh⁡(2κa)−cosh⁡(2κa)]C+=12sinh⁡(2κa)(κik−ikκ)C+=−i2sinh⁡(2κa)(κk+kκ)C+=−iξsinh⁡(2κa)C+ \begin{align*} A_{-} &= \frac{1}{2} \left[ \cosh (2\kappa a) +\frac{\kappa}{ik}\sinh (2\kappa a) -\frac{ik}{\kappa} \sinh (2\kappa a)-\cosh (2\kappa a) \right]C_{+} \\ &= \dfrac{1}{2} \sinh (2\kappa a) \left( \dfrac{\kappa}{ik}-\dfrac{ik}{\kappa} \right) C_+ \\ &= \dfrac{-i}{2} \sinh (2\kappa a) \left( \dfrac{\kappa}{k} + \frac{k}{\kappa} \right) C_+ \\ &= -i\xi \sinh (2\kappa a) C_{+} \end{align*}

This time, it was substituted by κk+kκ=2ξ\dfrac{\kappa}{k}+\dfrac{k}{\kappa}=2\xi. Therefore,

A−A+=−iξsinh⁡(2κa)C+ei2ka(cosh⁡(2κa)+iηsinh⁡(2κa))C+=−iξsinh⁡(2κa)e−i2kacosh⁡(2κa)+iηsinh⁡(2κa) \dfrac{A_{-}}{A_{+}} = \dfrac{-i\xi \sinh (2\kappa a) C_{+} } {e^{i2ka} \Big( \cosh (2\kappa a ) + i\eta \sinh (2\kappa a) \Big) C_+} = \dfrac{-i\xi \sinh (2\kappa a) e^{-i2ka} } { \cosh (2\kappa a ) + i\eta \sinh (2\kappa a)}

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