The entire information of a quantum system can be gathered with the help of(a) position (b) eigen value(c) momentum operator (d) wave function

States the information given byfour quantum numbers

The basic unit of data used in quantum computers

Using the expectation values from the previous questions, show that the eigenfunctions for the particle in a box obey theHeisenberg uncertainty for position and momentum, where we define the uncertainties statistically using standarddeviations as follows:  222 2 2222 2 2( )* ] ( ) ( )* ( )( )*[ ] (]) ( )*[ ] (ˆ ˆ[ [ˆ ˆ )xpx x x xx x xx x x dx x dxp p p d xx dxp           (a)And writing the Heisenberg uncertainty for position and momentum as2x x px p    Note that you need the square roots of the quantities defined in (a).Show that1/2 1/22 2a 2 2 1.136 / 22 3 a 2 3 2x x px p                             ħħħħ[5 marks]

A particle in a one dimensional infinite square well (box) is in a superposition of two quantum states:1 21234   where1 2and  are orthogonal, normalized (ortho-normal) eigenfunctions of the Hamiltonian.(a) Calculate the new normalization constants for1 2and  . [2 marks](b) A measurement of energy is made. What is the probability that the measurement will find energy1E ?What is the probability that the measurement will find energy2E ? [2 marks](c) Write down the integral expression for the evaluation of the expectation value of position,x - evaluateany “trivial” integrals, but leave as integrals any that require a full solution. [2 marks](d) Write down the integral expression for the evaluation of the expectation value of momentum,xp -evaluate any “trivial” integrals, but leave as integrals any that require a full solution

Set-up and solve the quantum mechanical problem of a particle of mass m insidea one-dimensional infinite potential well of width a. Find the energy eigen values andthe eigen wave functions. Show that the energy states are discrete. Sketch the wave-function, probability density and energy for first and second excited states of theparticle