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### $$\S\S$$ 7.14 Q[5] $$\int_0^\infty \frac{x^3}{e^x-1}=\frac{\pi^4}{15}$$

Summary (or abstract) not available for this node.
kapoor 28/10/18

### $$\S\S$$ 7.14 Integrals from Statistical Mechanics

Summary (or abstract) not available for this node.
kapoor 28/10/18

### Problem-QM-20020

Prove that $$\sigma_j \sigma_k = \delta_{jk} + i\epsilon_{jkl}\sigma_l$$ where $$\sigma_k, k=1,2,3$$ are Pauli matrices.

kapoor 28/10/18

### Problem-QM-20019

Show that  a  $$2\times 2$$ complex matrix, which anticommutes with all the three Pauli matrices, must be null matrix.

kapoor 28/10/18

### Problem-QM-20018

Is it correct to say that the Pauli matrices are
(a) hermitian   (b) unitary  (c)  idempotent   (d)  projection

kapoor 28/10/18

### Problem-QM-20017

If $\vec{\alpha}$ is a vector $(\alpha_1,\alpha_2,\alpha_3)$, show that  $$(\vec{\alpha}\cdot\vec{\sigma})^2 = |\vec{\alpha}|^2$$  where $$|\alpha|=\sqrt{\alpha_1^2+\alpha_2^2+\alpha_3^2}$$  Use this result and show that $\exp(i\vec{\alpha}\cdot\vec{\sigma}) = \cos{|\vec{\alpha}|} + i\vec{\alpha}\cdot\sigma \sin|\vec{\alpha}|$

kapoor 28/10/18

### Problem-QM-20016

Show that the Pauli matrices anti-commute. $$\sigma_j\sigma_k + \sigma_k\sigma_j =2\delta_{jk}$$ and that the square of each Pauli matrix is identity matrix. $$\sigma_k^2=I, \qquad k=1,2,3$$

kapoor 28/10/18

### Problem-QM-20013

Show that for a system of two identical particles having spin $s$, the ratio of the number of states, symmetric under exchange of spins, to the number of the antisymmetric states is given by ${(s+1)\over s}.$

kapoor 28/10/18

### QM-20 :: Spin and Identical Particles

Summary (or abstract) not available for this node.
kapoor 28/10/18

### Problem-QM-20015

Prove that the Pauli matrices satisfy the commutation relations.
$$[\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k$$

kapoor 28/10/18