Browse & Filter

The vector space needed to describe a particular physical system is two dimensional complex vector space. The  states are therefore represented by 2 component complex column vectors. The observables for this system are $2\times2$
matrices. A set of three dynamical variables of the system, $X,Y,Z,$ are be represented by $2\times 2$ matrices denoted by $\sigma_x, \sigma_y,\sigma_z$,  where $$  \sigma_x =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad \sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad \sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$ State which of the following operators can represent an observable quantity and which ones cannot represent an observable.

1.  $X_P = X+iY$
2.  $X_M=X-iY$ 
3.  $R=3X + 12Y + 4Z$
4.  $T= X^2+Y^2+Z^2$

Using spectral theorem, or otherwise, show that \begin{equation} \exp(i\hat{P}a) (\hat{x})^n \exp(-i\hat{P}a ) = (\hat{x}+a)^n, \end{equation} where \(n\) is an integer and  \(\hat{P}\) and \(\hat{x}\) are the momentum  and  position operators. 

A dynamical variable \(S\) of a system is represented by a \(3\times3\) matrix: \begin{eqnarray} S= \begin{pmatrix} 0 & 1 & 0\\ 1 & 0 & 1\\ 0&1&0 \end{pmatrix}. \end{eqnarray} 

1. Find all allowed values of \(S\). If the state of a system is given by the vector \[ f= \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} \]
2. Compute the probability of getting possible different values when a measurement of \(S\) is made.

If \(\psi(x,y,z)\)is the wave function of a particle in three dimension   write expression for probability that

1. \(x\) is between \(x_1\) and \(x_2\);
2. \(p_z\) lies between \(p_1\) and \(p_2\);
3. Assume that \(\psi(\vec{r}) =  R(r) Y_{\ell m}(\theta,\phi)\). What is the probability that the particle will be found inside a sphere of radius \(R\)? What is the probability that the particle will be found outside the sphere of radius \(R\)?

A particular state of a physical system is represented by \[\psi = \begin{pmatrix}  1 \\ 1+i \\ 1-i \end{pmatrix}.\] Find value of \(\lambda\) such that the vector \(\phi\), where \[ \phi= \begin{pmatrix} \lambda \\ 2i \\ 2\end{pmatrix}\] may represent the same state as \(\psi\).
Answer: \(\lambda = (1+i)\)