B.2 Chapters 3 and 4

B.2 Chapters 3 and 4

Concept links · terms present in this machine draft; source roles are unverified: complex inner product · eigenvalue · Lie algebra

Problem 1:

Calculate the exponential $e^{tM}$ for

$$\left(\begin{array}{ccc}0& \pi & 0\\ -\pi & 0& 0\\ 0& 0& 0\end{array}\right)$$

by two diferent methods:

• Diagonalize the matrix M (i.e., write as $PD{P}^{-1}$ , for $D$ diagonal), then show that

$$e^{tPD{P}^{-1}}=P{e}^{tD}{P}^{-1}$$

and use this to compute $e^{tM}$

• Calculate $e^{tM}$ using the Taylor series expansion for the exponential, as well as the series expansions for the sine and cosine.

Problem 2:

Consider a two-state quantum system, with Hamiltonian

$$H=-B_x{\sigma }_1$$

(this is the Hamiltonian for a spin $\frac{1}{2}$ system subjected to a magnetic field in the x-direction).

• Find the eigenvectors and eigenvalues of H. What are the possible energies that can occur in this quantum system?

• If the system starts out at time $t=0$ in the state

$$|\psi (0)⟩=\left(\genfrac{}{}{0ex}{}{1}{0}\right)$$

$(i.e.,$ spin ${{}^{\ast }\mathrm{up}}^{\mathfrak{V}})$ find the state at later times.

Problem 3:

By using the fact that any unitary matrix can be diagonalized by conjugation by a unitary matrix, show that all unitary matrices can be written as $e^X$ , for X a skew-adjoint matrix in $\mathfrak{u}(n)$

By contrast, show that

$$A=\left(\begin{array}{cc}-1& 1\\ 0& -1\end{array}\right)$$

is in the group $SL(2,\mathbf{C})$ , but is not of the form $e^X$ for any $X\in \mathfrak{sl}(2,\mathbf{C})$ (this Lie algebra is all 2 by 2 matrices with trace zero).

Hint: For 2 by 2 matrices $X$, one can show (this is the Cayley-Hamilton theorem: matrices $X$ satisfy their own characteristic equation det $(\lambda 1-X)=0$ ， and for 2 by 2 matrices, this equation is ${\lambda }^2-tr(X)\lambda +\det (X)=0)$

$$X^2-tr(X)X+\det (X)1=0$$

For $X\in \mathfrak{sl}(2,\mathbf{C}),tr(X)=0$ , so here $X^2=-\det (X)1$ . Use this to show that

$$e^X=\cos (\sqrt{\det (X)})1+\frac{\sin (\sqrt{\det (X)})}{\sqrt{\det (X)}}X$$

Try to use this for $e^X=A$ and derive a contradiction (taking the trace of the equation, what is cos $(\sqrt{\det (X)})?)$

Problem 4:

• Show that $M$ is an orthogonal matrix if its rows are orthonormal vectors for the standard inner product (this is also true for the columns).

• Show that $M$ is a unitary matrix if its columns are orthonormal vectors for the standard Hermitian inner product (this is also true for the rows).

Previous section: B.1 Chapters 1 and 2 · Next section: B.3 Chapters 5 to 7 · 中文版本

来源与版本

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

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：7f317d1896fa748af4cb9570310ed7801a8bef7ef97dd5d1d2ea35f0099d5952

OCR 产物 SHA-256：7f317d1896fa748af4cb9570310ed7801a8bef7ef97dd5d1d2ea35f0099d5952