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For the energy level the probability density for a particle of mass m is a one dimensional potential box of width L is given by
Given energy level E =
So, the given energy level corresponds to n = 3
No, of nodes in ψ_{3 }= 3  1 = 2
Probabilty density
πboncl length between CC atoms. f.i and v is calculated by using the empirical formula,
R_{μv }= 1.52Å  0.186 Å P_{μv}
where P_{μv}= is π  bond order between μ_{ }and v C atom .
According to HM O theory. delocalization energy o f Benzene is 2β. Then Tbond length in Benzene is
Since, delocalization energy = 2β
So, one extra delocalizedTbond Benzene.
The total energy of πbond = 4(According to HMO Theory)
Since, there are 6(C C ) bond in benzene.
So. each (CC) bond has 4/6 πbond = 2/s = 0.66
Now. by using formula,
R_{μv }= 1.52Å  0.186 Å x 0.66 = 1.39 Å
According to Huckel molecular orbital theory, the total energy of hypothetical localized Benzene (C_{6}H_{6}) molecule is
So, there are three localized ttbond. Ech have energy = 2α+ 2β
So, total localized energy= 3(2α + 2β) = 6α +6β
A perturbation is introduced in a box of width 1 unit, where V_{0} is ( << 1 ) is a positive constant. The first order energy correction to the first exited state is α V_{0} . The value of α is_________
First excited state wavefimction
First order eneigy correction to the first excited state is
σ_{x }, σy and σ_{z} are the Pauli m atrices. The expression is equal to 2σ_{x }σ_{y }+ σ_{y }σ_{x }is equal to
2σ_{x }σ_{y }+ σ_{y }σ_{x }
=_{ }σ_{x }σ_{y }+_{ x }σ_{y} + σ_{y }σ_{x }
= σ_{x }σ_{y}, as σ_{x }σ_{y }+ σ_{y }σ_{x }= 0
= iσz
For linear momentum vector operator and position vector operator the operator is equivalent to
The degeneracy of the n = 2 level for a three dimensional isotropic oscillator is ________. (an integer)
Degeneracy for a three dimensional isotropic oscillator g_{n} =
n= 2
The operation of the commutator on a function f ( x ) is equal to
The degeneracy of a quantum particle in a cubic box having energy three times that of the lowest energy is ________
Energy eigenvalue of a particle in a cubic box.
Groimd state energy corresponds to n_{x} = n_{y} = n_{z }= 1
Three times the ground state energy =
Consider a normalized molecular orbital constructed from two different atomic orbitals and that fonn an orthonormal set. Here i = √1. The value of c^{2} is ______ (upto one decim al place)
Given:
Nonnalization condition:
The normalized wavefiinctions ψ_{1} and ψ_{2} correspond to the ground state and the first excited states of a particle in a potential. The operator A acts on the wavefiinctions as The expectation value of the operator A for the state is ________. (upto two decimal places)
Three identical non interacting particles, each of spin 1/2 and mass m. are moving in a onedimensional infinite potential well given by
the energy of the lowest energy state of the system is:
Figure shows the configuration of the lowest energy state of the system
A certain 2level system lias stationary state energy with normalized wave functions
respectively. A small perturbation V is applied on the system such that V_{12} = 2. The wavefunction of the particle in groimd state collected upto first order, will be
Fii'st order collection in the ground state wave function,
Collected ground state wave function,
Given that work function of a metal is 5 eV and that photons of 9 eV are incident on it. the maximum velocity of emitted electrons is approximately______ x10^{6} m/s (upto one decimal place)
Maximum kinetic energy o qf the electrons,
The ground state wave function of hydrogen atom is given by where a_{0} is the Bohr
radius. The plot of the radial probability density P(r) for the hydrogen atom in the ground state is
The radial probability density is given by
Only option (d) satisfies this condition.
A particle of mass m is constrained to move in a circular ring of radius R . When a perturbation
(where a is a real constant) is added, the first order collection to the ground state energy is The value
of η is ______ (up to one decimal place)
Wavefiinction for a particle in a circular ring of radius R is given by
Ground state wave function corresponds to m = 0,
First order collection to ground state energy
The wavefiuiction for a linear harmonic oscillator described by N where α and N are constants, has
For small values of
For large values of
The wavefunction will cut yaxis aty = 2.
ψ is even function. so it is symetric about yaxis.
From the graph, there are two maxima and one minimum only.
The ‘‘trial'’ wavefuction for He atom is taken as where N is the normalization factor and Z ' is the variational parameter. Expression for the energy in terms of Z ' is found to be
Then the lowest value of the energy ( E ') for He atom i s _________________(in a.u .)
(Round off to two decimal places)
For minimmn energy,
The minimum energy will be
The average distance between the electron and nucleus of a He^{+} (singly ionised Heliumion) in the 1s state is
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