B.1 Chapters 1 and 2

B.1 Chapters 1 and 2

Concept links · terms present in this machine draft; source roles are unverified: vector space · irreducible representation · complexification

Problem 1:

Consider the group $S_3$ of permutations of 3 objects. This group acts on the set of 3 elements. Consider the representation $(\pi ,{\mathbf{C}}^3)$ this gives on the vector space ${\mathbf{C}}^3$ of complex valued functions on the set of 3 elements (as defined in section 1.3.2). Choose a basis of this set of functions, and find the matrices $\pi (g)$ for each element $g\in S_3$

Is this representation irreducible? If not, can you give its decomposition into irreducibles, and find a basis in which the representation matrices are block diagonal?

Problem 2:

Use a similar argument to that of theorem 2.3 for $G=U(1)$ to classify the irreducible diferentiable representations of the group $R$ under the group law of addition. Which of these are unitary?

Problem 3:

Consider the group $SO(2)$ of 2 by 2 real orthogonal matrices of determinant one. What are the complex irreducible representations of this group? (A hint: how are $SO(2)$ and $U(1)$ related?)

There is an obvious representation of $SO(2)$ on ${\mathbf{R}}^2$ given by matrix multiplication on real 2-vectors. If you replace the real 2-vectors by complex 2-vectors, but use the same representation matrices, you get a 2-complex dimensional representation (this is called “complexification”). How does this decompose as a direct sum of irreducibles?

Problem 4:

Consider a quantum mechanical system with state space $\mathcal{H}={\mathbf{C}}^3$ and Hamiltonian operator

$$H=\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 2\end{array}\right)$$

Solve the Schr¨odinger equation for this system to find its state vector $|\Psi (t)⟩$ at any time $t>0$ , given that the state vector at $t=0$ was

$$\left(\begin{array}{c}{\psi }_1\\ {\psi }_2\\ {\psi }_3\end{array}\right)$$

with ${\psi }_i\in \mathbf{C}$

Chapter contents · Next section: B.2 Chapters 3 and 4 · 中文版本

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正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 537、538、539、540、541、542、543、544、545、546、547、548、549、550、551、552、553、554、555、556

来源版本：2025-10-20

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