mathematics of quantum mechanics

合集 - 线代(6)

1.线性代数 linear algebra01-032.Introduction to Linear Algebra02-053.行列式2024-12-144.矩阵2024-12-31

5.mathematics of quantum mechanics02-07

6.Mathematics of Quantum Mechanics ii02-09

收起

NOTE

This is a summary of the book "Mathematics of Quantum Mechanics".

Part 1: Complex Numbers

1.1 What is a Complex Number?

• Imaginary Unit: Defined as $i=\sqrt{-1}$.
• Imaginary Numbers: Numbers whose square is negative.
• Complex Numbers: Numbers of the form $z = a+bi$, where $a$ and $b$ are real numbers. Here, $a$ is the real part, and $b$ is the imaginary part.
• Common Conclusions:
  • The modulus of a complex number $z=a + bi$ is always a non - negative real number: $\vert z\vert=\sqrt{a^{2}+b^{2}}\geq0$.
  • The conjugate of a complex number $z = a + bi$ is $\overline{z}=a - bi$, and $z\cdot\overline{z}=\vert z\vert^{2}$.

1.2 Operations with Complex Numbers

• Addition: If $z = a+bi$ and $w = c + di$, then $z + w=(a + c)+(b + d)i$.
• Multiplication: $zw=(ac - bd)+(ad + bc)i$.
• Conjugate: $\overline{z}=a - bi$.
• Modulus: $\vert z\vert=\sqrt{a^{2}+b^{2}}$.
• Common Conclusions:
  • The addition and multiplication of complex numbers are commutative and associative.
  • The modulus of a complex number is invariant under conjugation: $\vert\overline{z}\vert=\vert z\vert$.

1.3 Euler’s Formula and Polar Form

• Euler’s Formula: $e^{i\theta}=\cos\theta + i\sin\theta$.
• Polar Form: $z=\vert z\vert e^{i\theta}$.
• Common Conclusions:
  • Euler’s formula provides a way to represent complex numbers in polar form, which simplifies multiplication and division.
  • The polar form of a complex number highlights the magnitude and direction (angle) of the number in the complex plane.

Part 2: Linear Algebra

2.1 Vectors

• Vector Definition: A column or row of numbers.
• Vector Addition: If $\vec{v}=\begin{pmatrix}v_1\\v_2\\\vdots\\v_n\end{pmatrix}$ and $\vec{w}=\begin{pmatrix}w_1\\w_2\\\vdots\\w_n\end{pmatrix}$, then $\vec{v}+\vec{w}=\begin{pmatrix}v_1 + w_1\\v_2 + w_2\\\vdots\\v_n + w_n\end{pmatrix}$.
• Scalar Multiplication: If $c$ is a scalar and $\vec{v}=\begin{pmatrix}v_1\\v_2\\\vdots\\v_n\end{pmatrix}$, then $c\vec{v}=\begin{pmatrix}cv_1\\cv_2\\\vdots\\cv_n\end{pmatrix}$.
• Common Conclusions:
  • Vectors can be added and scaled, and these operations follow the commutative and associative properties.
  • The zero vector $\vec{0}$ acts as the additive identity in vector spaces.

2.2 Matrices

• Matrix Definition: A rectangular array of numbers.
• Matrix Addition and Scalar Multiplication: If $M=(M_{ij})$ and $N=(N_{ij})$, then $(M + N)_{ij}=M_{ij}+N_{ij}$; if $c$ is a scalar, then $(cM)_{ij}=c(M_{ij})$.
• Matrix Multiplication: If $M=(M_{ij})$ and $N=(N_{ij})$, then $(MN)_{ij}=\sum_{k}M_{ik}N_{kj}$.
• Transpose: $(M^{T})_{ij}=M_{ji}$.
• Conjugate Transpose: $(M^{\dagger})_{ij}=\overline{M_{ji}}$.
• Common Conclusions:
  • Matrix multiplication is associative but not commutative in general.
  • The identity matrix $I$ acts as the multiplicative identity for matrices.

2.3 Complex Conjugate, Transpose, and Conjugate Transpose

• Complex Conjugate of a Vector/Matrix:
  • For a vector $\vec{v}=\begin{pmatrix}v_1\\v_2\\\vdots\\v_n\end{pmatrix}$, its complex conjugate is $\vec{v}^*=\begin{pmatrix}v_1^*\\v_2^*\\\vdots\\v_n^*\end{pmatrix}$.
  • For a matrix $M=\begin{pmatrix}m_{11}&m_{12}&\cdots&m_{1n}\\m_{21}&m_{22}&\cdots&m_{2n}\\\vdots&\vdots&\ddots&\vdots\\m_{m1}&m_{m2}&\cdots&m_{mn}\end{pmatrix}$, its complex conjugate is $M^*=\begin{pmatrix}m_{11}^*&m_{12}^*&\cdots&m_{1n}^*\\m_{21}^*&m_{22}^*&\cdots&m_{2n}^*\\\vdots&\vdots&\ddots&\vdots\\m_{m1}^*&m_{m2}^*&\cdots&m_{mn}^*\end{pmatrix}$.
• Transpose of a Vector/Matrix:
  • For a vector $\vec{v}=\begin{pmatrix}v_1\\v_2\\\vdots\\v_n\end{pmatrix}$, its transpose is $\vec{v}^T=[v_1\ v_2\ \cdots\ v_n]$.
  • For a matrix $M=\begin{pmatrix}m_{11}&m_{12}&\cdots&m_{1n}\\m_{21}&m_{22}&\cdots&m_{2n}\\\vdots&\vdots&\ddots&\vdots\\m_{m1}&m_{m2}&\cdots&m_{mn}\end{pmatrix}$, its transpose is $M^T=\begin{pmatrix}m_{11}&m_{21}&\cdots&m_{m1}\\m_{12}&m_{22}&\cdots&m_{m2}\\\vdots&\vdots&\ddots&\vdots\\m_{1n}&m_{2n}&\cdots&m_{mn}\end{pmatrix}$.
• Conjugate Transpose (Adjoint) of a Vector/Matrix:
  • For a vector $\vec{v}=\begin{pmatrix}v_1\\v_2\\\vdots\\v_n\end{pmatrix}$, its conjugate transpose is $\vec{v}^\dagger=[v_1^*\ v_2^*\ \cdots\ v_n^*]$.
  • For a matrix $M=\begin{pmatrix}m_{11}&m_{12}&\cdots&m_{1n}\\m_{21}&m_{22}&\cdots&m_{2n}\\\vdots&\vdots&\ddots&\vdots\\m_{m1}&m_{m2}&\cdots&m_{mn}\end{pmatrix}$, its conjugate transpose is $M^\dagger=\begin{pmatrix}m_{11}^*&m_{12}^*&\cdots&m_{1n}^*\\m_{21}^*&m_{22}^*&\cdots&m_{2n}^*\\\vdots&\vdots&\ddots&\vdots\\m_{m1}^*&m_{m2}^*&\cdots&m_{mn}^*\end{pmatrix}$.
• Common Conclusions:
  • Taking the complex conjugate of a vector or matrix involves taking the complex conjugate of each of its elements. That is, for a vector $\vec{v}$, $(\vec{v}^*)_i = v_i^*$; for a matrix $M$, $(M^*)_{ij}=m_{ij}^*$.
  • Transposing a vector or matrix involves flipping the elements across its diagonal. For a vector $\vec{v}$, $(\vec{v}^T)_j = v_j$; for a matrix $M$, $(M^T)_{ij}=M_{ji}$.
  • The conjugate transpose (or adjoint) of a vector or matrix involves both taking the complex conjugate and transposing the elements. For a vector $\vec{v}$, $(\vec{v}^\dagger)_j = v_j^*$; for a matrix $M$, $(M^\dagger)_{ij}=M_{ji}^*$.
  • For a matrix $M$, $(M^\dagger)^\dagger = M$, $(M^T)^T = M$, and $(M^*)^* = M$.
  • The conjugate transpose is particularly important in quantum mechanics and other fields where complex numbers are used, as it preserves the inner - product structure. For example, for two vectors $\vec{u}$ and $\vec{v}$, $\langle\vec{u},\vec{v}\rangle=\langle\vec{v},\vec{u}\rangle^*$, and $\langle M\vec{u},\vec{v}\rangle=\langle\vec{u},M^\dagger\vec{v}\rangle$.

2.4 Inner Product and Norms

• Inner Product: If $\vec{v}=\begin{pmatrix}v_1\\v_2\\\vdots\\v_n\end{pmatrix}$ and $\vec{w}=\begin{pmatrix}w_1\\w_2\\\vdots\\w_n\end{pmatrix}$, then $\vec{v}\cdot\vec{w}=\sum_{i}v_i\overline{w_i}$.
• Norm: $\|\vec{v}\|=\sqrt{\vec{v}\cdot\vec{v}}$.
• Common Conclusions:
  • The inner product of two vectors is a scalar, and it measures the cosine of the angle between them.
  • The norm of a vector is a measure of its length, and it is always non - negative.

2.5 Basis

• Basis Definition: A set of linearly independent vectors that span the vector space.
• Orthogonal Basis: Basis vectors are orthogonal to each other.
• Standard Basis: An orthogonal basis where each basis vector has a norm of 1.
• Common Conclusions:
  • Any vector in a vector space can be expressed as a linear combination of basis vectors.
  • An orthogonal basis simplifies calculations involving inner products and projections.

Appendix A: Properties of Complex Numbers and Exponential Functions

A.1 Properties of Complex Numbers: Proofs

• Commutativity of Addition: Let $z = a + bi$ and $w=c + di$, where $a,b,c,d\in R$. Then $z + w=(a + c)+(b + d)i$ and $w + z=(c + a)+(d + b)i$. Since addition of real - numbers is commutative ($a + c=c + a$ and $b + d=d + b$), we have $z + w=w + z$.

• Commutativity of Multiplication: Let $z = a+bi$ and $w = c + di$. Then $zw=(a + bi)(c + di)=ac+adi + bci+bdi^{2}=(ac - bd)+(ad + bc)i$. And $wz=(c + di)(a + bi)=ca + cbi+adi+bdi^{2}=(ca - bd)+(cb + ad)i$. Since multiplication and addition of real - numbers are commutative ($ac = ca$, $bd = db$, $ad = da$, $bc = cb$), we have $zw = wz$.

• Modulus of a Complex Number: Given $z=a + bi$, $\vert z\vert=\sqrt{a^{2}+b^{2}}$. Since $a^{2}\geq0$ and $b^{2}\geq0$ for all real numbers $a$ and $b$, $a^{2}+b^{2}\geq0$, and $\sqrt{a^{2}+b^{2}}$ is a non - negative real number.

• Conjugate of a Complex Number: Given $z=a + bi$, $\overline{z}=a - bi$. Then $z\cdot\overline{z}=(a + bi)(a - bi)=a^{2}-(bi)^{2}=a^{2}+b^{2}=\vert z\vert^{2}$.

• Modulus Invariance under Conjugation: Given $z=a + bi$, $\overline{z}=a - bi$. Then $\vert\overline{z}\vert=\sqrt{a^{2}+(-b)^{2}}=\sqrt{a^{2}+b^{2}}=\vert z\vert$.

• Modulus of a Product: Let $z=a + bi$ and $w=c + di$. Then $zw=(ac - bd)+(ad + bc)i$. $\vert zw\vert=\sqrt{(ac - bd)^{2}+(ad + bc)^{2}}=\sqrt{a^{2}c^{2}-2acbd + b^{2}d^{2}+a^{2}d^{2}+2adbc + b^{2}c^{2}}=\sqrt{(a^{2}+b^{2})(c^{2}+d^{2})}=\vert z\vert\vert w\vert$.

-Triangle Inequality: Let $z=a + bi$ and $w=c + di$. Then $z + w=(a + c)+(b + d)i$. $\vert z + w\vert^{2}=(a + c)^{2}+(b + d)^{2}=a^{2}+2ac + c^{2}+b^{2}+2bd + d^{2}$. $(\vert z\vert+\vert w\vert)^{2}=\vert z\vert^{2}+2\vert z\vert\vert w\vert+\vert w\vert^{2}=a^{2}+b^{2}+2\sqrt{(a^{2}+b^{2})(c^{2}+d^{2})}+c^{2}+d^{2}$. By the Cauchy - Schwarz inequality $(ac + bd)^{2}\leq(a^{2}+b^{2})(c^{2}+d^{2})$, we have $\vert z + w\vert\leq\vert z\vert+\vert w\vert$.

• Inverse of a Complex Number: Given $z\neq0$, let $z=a + bi$. We want to find $z^{-1}$ such that $z\cdot z^{-1}=1$. Let $z^{-1}=x+yi$, then $(a + bi)(x + yi)=1$. Expanding gives $(ax - by)+(ay + bx)i = 1$. Solving the system $\begin{cases}ax - by = 1\\ay+bx = 0\end{cases}$, we get $x=\frac{a}{a^{2}+b^{2}}$ and $y =-\frac{b}{a^{2}+b^{2}}$. So $z^{-1}=\frac{\overline{z}}{\vert z\vert^{2}}$.

A.2 Euler’s Number and Exponential Functions

• Euler’s Number $e$: $e\approx2.718281828459045$.

• Exponential Function: For any real or complex number $x$, $e^{x}=\sum_{n = 0}^{\infty}\frac{x^{n}}{n!}$.

• Properties of Exponential Functions:

  -$e^{x + y}=e^{x}e^{y}$: $e^{x}=\sum_{n = 0}^{\infty}\frac{x^{n}}{n!}$, $e^{y}=\sum_{m = 0}^{\infty}\frac{y^{m}}{m!}$. Then $e^{x}e^{y}=\sum_{n = 0}^{\infty}\sum_{m = 0}^{\infty}\frac{x^{n}y^{m}}{n!m!}$. Let $k=n + m$, then $e^{x}e^{y}=\sum_{k = 0}^{\infty}\sum_{n = 0}^{k}\frac{x^{n}y^{k - n}}{n!(k - n)!}$. By the binomial theorem $\sum_{n = 0}^{k}\frac{x^{n}y^{k - n}}{n!(k - n)!}=\frac{(x + y)^{k}}{k!}$, so $e^{x}e^{y}=e^{x + y}$.

• $e^{xy}=(e^{x})^{y}$: Using the power - series definition and properties of exponents and series manipulation.

• $e^{-x}=\frac{1}{e^{x}}$: Since $e^{x}e^{-x}=e^{x+( - x)}=e^{0}=1$, then $e^{-x}=\frac{1}{e^{x}}$.

• $e^{0}=1$: Substitute $x = 0$ into $e^{x}=\sum_{n = 0}^{\infty}\frac{x^{n}}{n!}$, we get $e^{0}=\frac{0^{0}}{0!}+\sum_{n = 1}^{\infty}\frac{0^{n}}{n!}=1+0+\cdots = 1$.

A.3 Radians

• Definition of Radians: Given a circle of radius $r$, if an arc of length $l$ subtends an angle $\theta$ at the center of the circle, then $\theta=\frac{l}{r}$ (in radians).

• Conversion between Degrees and Radians: We know that a full - circle has an angle of $360^{\circ}$ and an arc - length equal to $2\pi r$. If $\theta_{degrees}$ is the angle in degrees and $\theta_{radians}$ is the angle in radians, then for a full - circle $\theta_{degrees}=360^{\circ}$ and $\theta_{radians}=2\pi$. So $\theta_{radians}=\theta_{degrees}\times\frac{\pi}{180}$.

Common Angles in Radians:

• $0^{\circ}=0\times\frac{\pi}{180}=0$ radians.
• $30^{\circ}=30\times\frac{\pi}{180}=\frac{\pi}{6}$ radians.
• $45^{\circ}=45\times\frac{\pi}{180}=\frac{\pi}{4}$ radians.
• $60^{\circ}=60\times\frac{\pi}{180}=\frac{\pi}{3}$ radians.
• $90^{\circ}=90\times\frac{\pi}{180}=\frac{\pi}{2}$ radians.
• $180^{\circ}=180\times\frac{\pi}{180}=\pi$ radians.

A.4 Proof of Euler’s Theorem

• Euler’s Formula: $e^{i\theta}=\cos\theta+i\sin\theta$.

• Proof Using Taylor Series:

  • The Taylor series for $e^{x}=\sum_{n = 0}^{\infty}\frac{x^{n}}{n!}=1 + x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots$.
  • Substitute $x = i\theta$ into the Taylor series of $e^{x}$: $e^{i\theta}=1 + i\theta+\frac{(i\theta)^{2}}{2!}+\frac{(i\theta)^{3}}{3!}+\frac{(i\theta)^{4}}{4!}+\frac{(i\theta)^{5}}{5!}+\cdots$.
  • Since $i^{2}=-1$, $i^{3}=i^{2}\cdot i=-i$, $i^{4}=(i^{2})^{2}=1$, $i^{5}=i^{4}\cdot i = i$, etc.
  • $e^{i\theta}=1 + i\theta-\frac{\theta^{2}}{2!}-\frac{i\theta^{3}}{3!}+\frac{\theta^{4}}{4!}+\frac{i\theta^{5}}{5!}-\cdots=(1-\frac{\theta^{2}}{2!}+\frac{\theta^{4}}{4!}-\cdots)+i(\theta-\frac{\theta^{3}}{3!}+\frac{\theta^{5}}{5!}-\cdots)$.
  • The Taylor series for $\cos\theta=\sum_{n = 0}^{\infty}(- 1)^{n}\frac{\theta^{2n}}{(2n)!}=1-\frac{\theta^{2}}{2!}+\frac{\theta^{4}}{4!}-\cdots$.
  • The Taylor series for $\sin\theta=\sum_{n = 0}^{\infty}(-1)^{n}\frac{\theta^{2n + 1}}{(2n+1)!}=\theta-\frac{\theta^{3}}{3!}+\frac{\theta^{5}}{5!}-\cdots$.
  • So $e^{i\theta}=\cos\theta+i\sin\theta$.