Browse & Filter

   For a particle having the wave function           $$\psi(x) = N
\exp(-x^2/\alpha^2)$$       compute the averages of the following dynamical
variables.
          (a) kinetic energy,
          (b) $V_1(x) = V_0 |x|^{2m+1}$
          (c)  $V_2(x) =kx^2$

For a harmonic oscillator in the ground state find the average values of
kinetic energy, potential energy and $|x|^{2m+1}.$

Given that :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 $$  \sigma_x =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad
\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$ Answer the following question
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Question :      What vector would represent the state of the system if it is known that
the system has definite value $+1$ for the dynamical variable $X$? What vector
would represent the state if the system has definite value $-1$ for the variable
$Y$.

 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  $$  \sigma_x
=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
\sigma_y=\begin{pmatrix} 0&-i\\i&0\end{pmatrix}, \qquad
\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$     What values are
experimentally allowed if one measures the dynamical variable

1.        $X$ 
2.       $Z$ 
3.       $T= X^2+Y^2+Z^2$

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 $$  \sigma_x =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad \sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad \sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$  If the state vector of the system is given by $$   \left( \begin{array}{r} \displaystyle{1\over\sqrt{5}}\\[4mm] \displaystyle {-2\over\sqrt{5}} \end{array} \right) $$ Find the probability that

1. a measurement of $X$ will give value $1$.
2. a measurement of $Y$ will give value $-1$.
3. a measurement of $Z$ will give value $1$.
4. a measurement of $X+Y$ will give value $\sqrt{2}$.

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 \(R\)?