Sakurai Definition and 79 Threads

In Modern Quantum Mechanics (2nd ed.) by J.J. Sakurai, in section 4.4 on 'The Time-Reversal Discrete Symmetry' he derives the time-reversal operator, \Theta, for the spin-$\frac{1}{2}$ case as (pg.: 277, eq. (4.4.65)): \Theta = \eta e^{\frac{-i \pi S_{y}}{\hbar}}K = -i \eta \left(...

The following is from 'Modern Quantum Mechanics' by J.J. Sakurai, page 159. \left( \frac{ \hbar }{ 2 } \right) \exp \left( \frac{i S_{z} \phi}{\hbar} \right) \left{ left( \mid + \rangle \langle - \mid \right) + \left( \mid - \rangle \langle + \mid \right) right} \exp \left( \frac{- i S_{z}...

Unknown "well-known" equation in Sakurai On p391 in Modern Quantum Mechanics is stated: Now we use the well-known equation: \frac{1}{E - H_0 -i \epsilon} = Pr. \left( \frac{1}{E-H_0} \right) + i \pi \delta \left(E-H_0 \right) Frankly I've never heard of the equation or the notation Pr. My...

Homework Statement I have attached a link to the solution set of some questions from sakurai. But I don't understand clearly the solution of question number 5.29. link: http://www.people.virginia.edu/~7Erdb2k/homework/phys752/quantum.pdf" Homework Equations how did they found the 4x4 matrix...

Homework Statement c) explicit calculations, using the usual rules of wave mechanics, show that the wave function for a Gaussian wave packet given by \left\langle {x'|\alpha } \right\rangle = {(2\pi {d^2})^{ - 1/4}}\exp \left( {{\bf{i}}{\textstyle{{\left\langle p \right\rangle x'} \over...

Homework Statement Suppose that |i> and |j> are eigenkets of some Hermitian operator. Under what condition can we conclude that |i> + |j> is also an eigenket of A? Justify your answer.Homework Equations It seems that all that is needed is for "A" to be a linear operator and for |i> and |j> to...

Homework Statement Not relevant, but I have some work that reaches incorrect conclusions, and I can't see the mistake "in the middle". Homework Equations The Attempt at a Solution \begin{array}{l} {(XY)^\dag } = {\left( {\left\langle {a'} \right|X\left| {a''} \right\rangle...

Homework Statement You know {\rm{Tr}}(XY) = \limits^{?} {\rm{Tr}}(YX), but prove it, using the rules of bra-ket algebra, sucka. (The late Sakurai does not actually call his reader "sucka"). Homework Equations {\mathop{\rm Tr}\nolimits} (X) \equiv \sum\nolimits_{a'} {\left\langle {a'}...

Homework Statement This isn't a homework problem. I am reading Sakurai (Modern Quantum Mechanics) and came upon this:

Homework Statement A spin ½ system is known to be in an eigenstate of \textbf{S}\cdot\hat{\textbf{n}} with eigenvalue \frac{\hbar}{2} , where \hat{\textbf{n}} is a unit vector lying in the xy-plane that makes an angle γ with the positive z-axis. a. Suppose S_x is measured. What is the...

Ok, maybe the subject has been discussed in some manner in other posts, but I would like a clear comparison between : * David J Griffiths - Introduction to Quantum Mechanics * Shankar - Principles of Quantum Mechanics * Sakurai - Modern Quantum Mechanics What is your experience with these...

Hi, I was advised to learn QM from Sakurai since I was interested in learning QM. However, my university's library doesn't have a copy so I can't look through it to decide if it is suitable. I am familiar with all the basic linear algebra (orthogonality, diagonalisation, eigenvectors...

Problem Show for the one-dimensional simple harmonic oscillator \langle 0 | e^{ikx} | 0 \rangle = \exp{[-k^2 \langle 0 | x^2 | 0 \rangle / 2]} where x is the position operator (here, k is a number, not an operator, with dimensions 1/length). My Solution Well, I already know how to do this...

Problem 23: If a certain set of orthonormal kets, |1> |2> |3> , are used as the base kets, the operators A and B are represented by A = \left( \begin{array}{ccc} a & 0 & 0 \\ 0 & -a & 0 \\ 0 & 0 &...

Homework Statement Determine the eigenvector for (S \cdot \hat{n}) |eigenvector> = (\hbar)/2 |eigenvector> where S = (\hbar)/2 \sigma. The sigmas are the Pauli spin matrices and \hat{n} = sin\beta cos\alpha \hat{i} + sin\beta\ sin\alpha \hat{j} + cos\beta \hat{k} You have to solve for the...

Problem Suppose that f(A) is a function of a Hermitian operator A with the property A|a'\rangle = a'|a'\rangle. Evaluate \langle b''|f(A)|b'\rangle when the transformation matrix from the a' basis to the b' basis is known.The attempt at a solution Here's what I have... I'm not sure if the last...

I'm pretty sure this is correct, but could someone verify for rigor? Problem Two observables A_1 and A_2, which do not involve time explicitly, are known not to commute, yet we also know that A_1 and A_2 both commute with the Hamiltonian. Prove that the energy eigenstates are, in general...

Homework Statement (Sakurai 1.27) [...] evaluate \langle \mathbf{p''} | F(r) | \mathbf{p'} \rangle Simplify your expression as far as you can. Note that r = \sqrt{x^2 + y^2 + z^2}, where x, y and z are operators. Homework Equations \langle \mathbf{x'} | \mathbf{p'} \rangle = \frac{1}{ {(2 \pi...

Homework Statement http://www.ocf.berkeley.edu/~yayhdapu/postings/sak19.gif Homework Equations The Attempt at a Solution http://www.ocf.berkeley.edu/~yayhdapu/postings/sakurai1.9.pdf" http://www.ocf.berkeley.edu/~yayhdapu/postings/sakurai1.9.docx" These are here in this attached...

Just out of curiosity, how hard is Fitzpatrick's online Graduate version of quantum mechanics compared to the Feynman lectures and Sakurai's book?

Homework Statement Using the orthonormality of |+\rangle and |-\rangle, prove [S_i,S_j]= i \varepsilon_{ijk}S_k where S_x = \frac{\hbar}{2}|+\rangle \langle - | + | - \rangle \langle + | S_y = -\frac{i\hbar}{2}|+\rangle \langle - | + | - \rangle \langle + | S_z =...

Evaluate \exp (i f(A)) in ket-bra form, where A is a Hermitian operator whose eigenvalues are known. \exp (i f(A)) = \exp(i f(\sum_i a_i \langle a_i |)). I'm a little bit stuck on where to go from here. Is f supposed to be a matrix values function of a matrix variable or what?

[SOLVED] Sakurai Ch.3 Pr.6 Homework Statement Let U = \text{e}^{i G_3 \alpha} \text{e}^{i G_2 \beta} \text{e}^{i G_3 \gamma}, where ( \alpha , \beta , \gamma ) are the Eulerian angles. In order that U represent a rotation ( \alpha , \beta , \gamma ) , what are the commutation rules...

Does someone know a site with Sakurai (Advanced QM) solved problems? Thanks in advance, Leticia

Homework Statement Consider a one-dim harm osc; start with the Schrödinger equation (SE) for the state vector, then derive the SE for the momentum-space wave function. The Attempt at a Solution My answer is this, all primed letters are numbers (as in sakurai notation). Its going to...

Of Modern Quantum Mechanics. This starts with a Hamiltonian H = a(|1\rangle\langle 1| - |2\rangle\langle 2| + |1\rangle\langle 2| + |2\rangle\langle 1|) This has eigenvalues \pm a\sqrt{2}. Shouldn't a Hamiltonian have only non-negative eigenvalues? If the sign in front of the...

Hello! I'm just doing the Problems of Chapter 1 of Sakurai: Modern Quantum Mechanics. On page 60, problem 2 he writes: "Suppose a 2x2 matrix X, (not necessary Hermitian, nor unitary) is written as X = a_0 + \mathbf{\sigma \cdot a} , where a_0 and a_{1,2,3} are numbers." which confuses...

Where can i find free ebooks about physics ? For example quantum mechanics by j.j.sakurai

I hope this is not off-topic here: I remember I had seen a time ago somewhere in internet the solutions to the excercises of Sakurai's book 'Modern Quantum Mechanics', but I am not able to find the link again. Anyone knows? Thanks.