Consider a two state system with normalized energy eigen state ψ1 & ψ2 and energy E1 < E2 what is the possible range for the expectation value of on an orbitrary linear combination of two state
(H) > E1 ......(1)
similarly
......(2)
form equation (1) & (2)
Suppose a wave function and an operator is given by
is given by
We have operator,
Now, my operator will be
∴
∴
∴
Consider the following Stem-Gerlach apparatus incoming beam consist of electron 2/3 of them having spin and other 1/3 have spin
in the z-direction
what fraction of the incident beam are detected in the up and down outputs of the apparatus
Consider a system whose initial state at r = 0 is given in term of a complete and orthonormal set of three vectors as follows
the probability of find tlie system at tune t in state
is _______ (upto two decunalplaces)
We have wave function at t = 0
Therefore, the wave function at t = t is given by
(where, E1, E2 and E3, are energy of difference states)
The probability of finding the system at time t in state
An election is confined in an infinite square well of width 10x10^-15 m. Calculate the wavelength of the electron emitted when the proton undergoes a transition from the first excited state (n=2) to the ground state (n=1).
The energy of the particle in the box of width L is given by
λ = 201 fm
Consider a tliree dimensional harmonic oscillator with Hamiltonian
The number of distinct eigenstates with energy eigenvalue 5/2 ℏω is_________
(Answer should be an integer).
Energy expression for 3-D harmonic oscillator,
The degneracyofnth state of an isolated 3D harmonic oscillator is given by
Consider an operator for a system of total angular momentum j = 1 then winch of the following state having non zero uncertainty
Operator Jx in matrix form for J = 1
The eigenvalue equation is given by
Consider is the eigen function of Jx corresponding the eigen value λ = 0
And according normalization condition,
⇒
∴
∴
Similarly the eigen function corresponding to others two eigen values are
As are the eigen state of operator Jx.
Therefore, uncertainty of Jx will be zero for these three states and is not an eigen states of Jx.
So, uncertainty of Jx will be non-zero for this state.
In the simple harmonic oscillator
what are the condition on m and m' for to be non-zero
For non zero of
⇒ m = n ± 1
And
Find the differential cross-section for die scattering of slow (low velocity) particle from a spherical delta poten- tial V (r) = V0 δ (r - a)
In case, the incident particle have low velocities only S-waves, L = 0 contributes in the scattering
Therefore, scattering amplitude is given by
Differential cross-section
For case of n = 2, ℓ = 1, m = 0 the value of r at which the radial probability density of the hydrogen atom reaches its maximum is ____ a0. (answer should be an integer)
Tlie probability density is given by
For the maximum of tlie probability density
Consider a system winch is mtially in the state was measured with value -ℏ, the probability is_______(upto one decimal place)
for eigenvalue mL = -1, eigenfunction probability of measuring Lz in
state is
An electron is confined in the ground state of a one dimensional harmonic oscillator such that energy required to excite to its first excited state is _____ (MeV)
The virial theorem states that
for ground state
for harmonic oscillator
Energy required to excite the electron to its first excited state
A particle of mass m coming in from the left with energy E > 0, encounters barrier potential
The wave function is given by
The constant A and B satisfy which one of the following relation?
The barrier potential is given as
The wave function is given as
Hie quantity B in equation (i) represent the amplitude ot reflected ray in medium 1 and A in equation (ii) represent the amplitude of transmitted ray in medium III
Hence, Reflectivity R = |A|2 and transmitivity T = | B |2
smce R + T = 1
⇒
Calculate the width of the probability density distribution for r(i.e Br) for hydrogen atom for the state
Tlie width of the probability distribution is given by
width ot probability distribution
At t = o, a state is given by
Where are ortlionomial stationary states of energy El & E2 respectively with E2 > El what is the shortest tune T > 0 for w hich
is orthogonal to
At t = 0, the wave function is given by
Therefore, the wave function at later time t = T is given by
These two states ortliogonal at time T
For shortest time m = 0
If the state of a particle moving in one dimensional harmonic oscillator is given by
Where represent the normalized nth energy eigenstate find the expectation value of number operator
A particle of mass m moves in a one dimensional potential box
Consider the V0 part as perturbation, using first order perturbation method calculate the energy of ground state.
Tlie energies and wavefunction of a particle in flat box of length 6a
energy correction to first order perturbation
A spin state precesses in a magnetic field same way as the classical magnetic dipole precesses in magnetic field with lasmor frequency given by consider the Hamiltonian
Larrnor frequency
is (in tem is of ω0)
(answer should b e an mteger)
From equation (i) and (ii)
Consider a system of four non-interacting identical spin 1/2 particles that are in same state and confined to move in a one-diniension infinite potential well of length a: V(x) = 0 for 0 < x < a and V(x) = ∞ forotlier values of x. The ground state energy of the system in units of
is (answer should be an integers).
The energy of the particle in one dimensional infinite potential box is given by
4 particles having same state
ground state energy'of system is
Consider a system whose intial state and hamiltonian are given by
find the total energy of a system
eigenvalues of energy
λ = -5,3, 5; E1 = - 5, E2 = 3, E3 = 5
and their correspondmg eigenvectors
expanding in terms of eigenvectors.
Total energy of the system
Second method:
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