Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

I. E. Irodov Solutions for Physics Class 11 & Class 12

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The document Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev is a part of the JEE Course I. E. Irodov Solutions for Physics Class 11 & Class 12.
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Q.66. Estimate the minimum errors in determining the velocity of an electron, a proton, and a ball of mass of 1 mg if the coordinates of the particles and of the centre of the ball are known with uncertainly 1µm.

Ans. From the uncertainty principle (Eqn. (6.2b))
Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

ThusIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

orIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

For an electron this means an uncertainty in velocity of 116 m/s if Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

For a proton          Δvx = 6.3 cm/s 

For a ball                Δvx = 1 x 10-20cm/s

 

Q.67. Employing the uncertainty principle, evaluate the indeterminancy of the velocity of an electron in a hydrogen atom if the size of the atom is assumed to be l = 0.10 nm. Compare the obtained magnitude with the velocity of an electron in the first Bohr orbit of the given atom.

Ans. As in the previous problem

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

The actual velocity v1 has been calculated in problem 6.21. It is

v= 2.21 x 106 m/s

Thus Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev (They are of the same order of magnitude)

 

Q.68. Show that for the particle whose coordinate uncertainty is Δx = λ/2π, where λ is its de Broglie wavelength, the velocity uncertainty is of the same order of magnitude as the particle's velocity itself.

Ans.  If  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Thus Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Thus Δv is of the same order as v.

 

Q.69. A free electron was initially confined within a region with linear dimensions l = 0.10 nm. Using the uncertainty principle, evaluate the time over which the width of the corresponding train of waves becomes η = 10 times as large. 

Ans.  Initial uncertainty  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev With this incertainty the wave train will spread out to a distance ηl long in time

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRevIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

 

Q.70. Employing the uncertainty principle, estimate the minimum kinetic energy of an electron confined within a region whose size is l = 0.20 nm.

Ans. Clearly  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Now  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev  and so

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Thus Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

 

Q.71. An electron with kinetic energy T ≈  4 eV is confined within a region whose linear dimension is l = 1 µm. Using the uncertainty principle, evaluate the relative uncertainty of its velocity. 

Ans. The momentum the electron is  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Uncertainty in its momentum is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Hence relative uncertainty

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Substitution gives

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

 

Q.72. An electron is located in a unidimensional square potential well with infinitely high walls. The width of the well is l. From the uncertainty principle estimate the force with which the electron possessing the minimum permitted energy acts on the walls of the well. 

Ans. By uncertainty principle, the uncertainty in momentum

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

For the ground state, we expect Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev so 

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

The force excerted on the wall can be obtained most simply from

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

 

Q.73. A particle of mass m moves in a unidimensional potential field U = kx2/2 (harmonic oscillator). Using the uncertainty principle, evaluate the minimum permitted energy of the particle in that field. 

Ans. We write

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

i.e. all four quantities are of the same order of magnitude. Then

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Thus we get an equilibrium situation (E = minimum) when

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

and then Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Quantum mechanics gives

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

 

Q.74. Making use of the uncertainty principle, evaluate the minimum permitted energy of an electron in a hydrogen atom and its corresponding apparent distance from the nucleus. 

Ans. Hence we write

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Then

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Hence Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev   for the equilibrium state.

and then Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

 

Q.75. A parallel stream of hydrogen atoms with velocity v = 600 m/s falls normally on a diaphragm with a narrow slit behind which a screen is placed at a distance l = 1.0 m. Using the uncertainty principle, evaluate the width of the slit S at which the width of its image on the screen is minimum. 

Ans. Suppose the width of the slit (its extension along they - axis) is δ. Then each electron has an uncertainty  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRevThis translates to an uncertainty

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev We must therefore have

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

For the image, hrodening has two sources.

We write

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

where Δ' is the width caused by the spreading of electrons due to their transverse momentum.
We have

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Thus

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

For large Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRevand quantum effect is unimportant. For small δ, quantum effects are large. But A (δ) is minimum when

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

as we see by completing the square. Substitution gives

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

 

Q.76. Find a particular solution of the time-dependent Schrodinger equation for a freely moving particle of mass m. 

Ans. The Schrodinger equation in one dimension for a free particle is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

we write Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev Then

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Then Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

E must be real and positive if φ(x) is to be bounded everywhere. Then

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

This particular solution describes plane waves.

 

Q.77. A particle in the ground state is located in a unidimensional square potential well of length 1 with absolutely impenetrable walls (0 < x < l). Find the probability of the particle staying within a region Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Ans.  We look for the solution of Schrodinger eqn. with

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev               (1)

The boundary condition of impenetrable walls means

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

The solution of (1) is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Then Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

A = 0 so

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Hence Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Thus the ground state wave function is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

We evaluate A by nomalization

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Thus Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Finally, the probability P for the particle to lie in Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

 

Q.78. A particle is located in a unidimensional square potential well with infinitely high walls. The width of the well is 1. Find the normalized wave functions of the stationary states of the particle, taking the midpoint of the well for the origin of the x coordinate. 

Ans. 

Here   Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev. Again wc have

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Then the boundary condition Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

gives   Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

There are two cases.

(1) Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

gives even solution. Here

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

and  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

n = 0 , 1 , 2 , 3 , . . .

This solution is even under x → - x .

(2) B = 0 , Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev This solution is odd.

 

Q.79. Demonstrate that the wave functions of the stationary states of a particle confined in a unidimensional potential well with infinitely high walls are orthogonal, i.e. they satisfy the condition 

 Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Here l is the width of the well, n are 
 integers.

Ans. The wave function is given in 6.77. We see that 

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

If n = n', this is zero as n and n' are integers.

 

Q.80. An electron is located in a unidimensional square potential well with infinitely high walls. The width of the well equal to 1 is such that the energy levels are very dense. Find the density of energy levels dN/dE, i.e. their number per unit energy interval, as a function of E. Calculate dN/dE for E = 1.0 eV if l = 1.0 cm. 

Ans. 

We have found that  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Let N (E) = number of states upto E. This number is n. The number of states upto E + dE is N( E + dE) = N(E ) + d N ( E ) . Then dN (E ) - 1 and

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

where ΔE = difference in energies between the nth & (n + 1)th level

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Thus

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

For the given case this gives Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev  levels per eV

 

Q.81. A particle of mass m is located in a two-dimensional square potential well with absolutely impenetrable walls. Find: 

(a) the particle's permitted energy values if the sides of the well are l1, and l2;
 (b) the energy values of the particle at the first four levels if the well has the shape of a square with side l.

Ans. 

(a) Here the schroditiger equation is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

we take the origin at one of the comers of the rectangle where the particle can lie. Then the wave function must vanish for

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

or      y = 0 or y = l2 .

we look for a solution in the form

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

cosines are not permitted by the boundary condition. Then

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

and Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Here n1, n2 are nonzero integers,

(b) If Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev then 

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

1st level :Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

2nd level :Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

3rd level :Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

4th level :Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

 

Q.82. A particle is located in a two-dimensional square potential well with absolutely impenetrable walls (0 < x < a, 0 < y < b).
 Find the probability of the particle with the lowest energy to be located within a region 0 < x < a/3. 

Ans. 

The wave function for the ground state is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

we find A by normalization

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Thus Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Then the requisite probability is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev on doing the y integral

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 2 Notes | EduRev

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