mathematics of quantum mechanics

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 \(e^{i\theta}=\cos\theta+i\sin\theta\).