Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

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Q.83. A particle of mass m is located in a three-dimensional cubic potential well with absolutely impenetrable walls. The side of the cube is equal to a. Find:
 (a) the proper values of energy of the particle;
 (b) the energy difference between the third and fourth levels;
 (c) the energy of the sixth level and the number of states (the degree of degeneracy) corresponding to that level. 

Ans. We proceed axactly as in (6.81). The wave function is chosen in the form 

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE = A sin k1x sin ky sin k3 z .

(The origin is at one comer of the box and the axes of coordinates are along the edges.) The boundary conditions are that Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE for

x = 0, x = a,  y = 0, y = a, z = 0, z = a

This gives

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The energy eigenvalues are

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The first level is (1, 1, 1). The second has (1, 1, 2), (1, 2, 1) & (2, 1, 1). The third level is (1, 2, 2) or (2, 1, 2) or (2, 2, 1). Its eneigy is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The fourth energy level is (1, 1, 3) or (1, 3, 1) or (3, 1, 1)

Its eneigy is Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(b) Thus Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(c) The fifth level is (2, 2, 2). The sixth level is (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)
Its eneigy is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

and its degree of degeneracy is 6 (six).

 

Q.84. Using the Schrodinger equation, demonstrate that at the point where the potential energy U (x) of a particle has a finite discontinuity, the wave function remains smooth, i.e. its first derivative with respect to the coordinate is continuous. 

Ans. We can for definiteness assume that the discontinuity occurs at the point x = 0. Now the schrodinger equation is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

We integrate this equation around x = 0 i.e., from Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE where Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE are small positive numbers. Then 

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

or  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Since the potential and the energy E are finite and ψ(x) is bounded by assumption, the integral on the right exists and Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Thus Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

So Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE is continuous at x = 0 (the point where U (x) has a finite jump discontinuity.)

 

Q.85. A particle of mass m is located in a unidimensional potential field U (x) whose shape is shown in Fig. 6.2, where U(0) = Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE. Find: 

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(a) the equation defining the possible values of energy of the particle in the region E < U0; reduce that equation to the form 

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

where Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEESolving this equation by graphical means, demonstrate that the possible values of energy of the particle form a discontinuous spectrum;
 (b) the minimum value of the quantity l2U0 at which the first energy level appears in the region E < U0. At what minimum value of l2Udoes the nth level appear?

Ans.

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(a) Starting from the Schrodinger equation in the regions l & II

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE   (1)

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE    (2)

where U0 > E > 0 , we easily derive the solutions in I & II

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE     (3)

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE        (4)

where Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The boundary conditions are

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE        (5)

andIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEare continuous at x = /, and ψ must vanish at x = + Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE.

ThenIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

andIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

so Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

From this we get

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

or Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE (6)

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Plotting the left and right sides of this equation we can find the points at which the straight lines cross the sine curve. The roots of the equation corresponding to the eigen values of energy Eand found from the inter section points (kl), for which tan (kl)i < 0 (i.e. 2nd & 4th and other even quadrants). It ii seen that bound states do not always exist. For the first bound state to appear (refer to the line (b) above)

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(b) Substituting, we getIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

as the condition for the appearance of the first bound state. The second bound state will appear when Id is in the fourth quadrant The magnitude of the slope of the straight line must then be less than

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Corresponding to Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

For n bound states, it is easy to convince one self that the slope of the appropriate straight line (upper or lower) must be less than

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

ThenIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Do not forget to note that for large n both + and - signs in the Eq. (6) contribute to solutions.

 

Q.86. Making use of the solution of the foregoing problem, determine the probability of the particle with energy E = U0/2 to be located in the region Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Ans.

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

and  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

or  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

It is easy to check that the condition of the boud state is satisfied. Also

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Then from the previous problem

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

By normalization

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The probability of the particle to be located in the region x > l is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

 

Q.87. Find the possible values of energy of a particle of mass m located in a spherically symmetrical potential well U (r) = 0 for r < r and U (r) = Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE for r = r0, in the case when the motion of the particle is described by a wave function ψ(r) depending only on r. 

Instruction. When solving the Schrodinger equation, make the substitution ψ(r) = x (r)/r. 

Ans. The Schrodinger equation is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

when ψ depends on r only,Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

If we put  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

and Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEThus we get

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The solution is Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

and   Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(For r < r0 we have rejected a term 5 cos k r as it does not vanish at r = 0). Continuity of the wavefunction at r = r0 requires

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Hence Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

 

Q.88. From the conditions of the foregoing problem find:
 (a) normalized eigenfunctions of the particle in the states for which ψ(r) depends only on r;
 (b) the most probable value rpr for the ground state of the particle and the probability of the particle to be in the region r < rpr.

 

Ans. (a) The nomalized wave functions are obtained from the normalization

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

HenceIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(b) The radial probability distribution function is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

For the ground state  n = 1

so Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

By inspection this is maximum for Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE. Thus Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The probability for the particle to be found in the region r < rpr is clearly 50 % as one can immediately see from a graph of sin2x.

 

Q.89. A particle of mass m is located in a spherically symmetrical potential well U (r) = 0 for r < r0  and U (r) = U0  for r >
 (a) By means of the substitution ψ(r) = x(r)/r find the equation defining the proper values of energy E of the particle for E < U0, when its motion is described by a wave function ψ(r) depending only on r. Reduce that equation to the form 

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(b) Calculate the value of the quantity Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEat which the first level appears. 

 

Ans.  If we put Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

the equation for X(r) has the from Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

which can be written as  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

andIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

where Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The boundary condition is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

These are exactly same as in the one dimensional problem in problem (6.85) Wc therefore omit further details
 

Q.90. The wave function of a particle of mass in in a unidimensional potential field U (x) = kx2/2 has in the ground state the form ψ(x) = Ae-αx2, where A is a normalization factor and a is a positive constant. Making use of the Schrodinger equation, find the constant a and the energy E of the particle in this state.

Ans.  The Schrodinger equation is Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

We are givenIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Then Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Substituting we find that following equation must hold

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

since Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE the bracket must vanish identicall. This means that the coefficient of x2 as well the term independent of x must vanish. We get

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Putting k/m = ω2, this leads to Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE
 

Q.91. Find the energy of an electron of a hydrogen atom in a stationary state for which the wave function takes the form ψ(r) = A (1 + ar) e-αr, where A, a, and α are constants. 

Ans.  The Schrodinger equation for the problem in Gaussian units

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

In MKS units we should read Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

we put Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE (1)

We are given that Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

so Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE   Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Equating the coefficients of r2, r, and constant term to zero we get

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE   (2)

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE     (3)

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE  (4)

From (3) either a = 0, Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

In the first case Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

This state is the ground state.

It n the second,  case Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

This state is one with n = 2 (2s).

 

Q.92. The wave function of an electron of a hydrogen atom in the ground state takes the form ψ(r) = Ae-r/r1, where A is a certain constant, r1 is the first Bohr radius. Find: (a) the most probable distance between the electron and the nucleus;
 (b) the mean value of modulus of the Coulomb force acting on the electron;
 (c) the mean value of the potential energy of the electron in the field of the nucleus. 

Ans. We first find A by normalization 

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

since the integral has the value 2.

Thus Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(a) The most probable distance rpr is that value of r for which

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

is maximum. This requires

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

or  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(b) The coulomb force being given by Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE , the mean value o f its modulus is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

In MKS units we should read  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(c)Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

In MKS units we should readIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

 

Q.93. Find the mean electrostatic potential produced by an electron in the centre of a hydrogen atom if the electron is in the ground state for which the wave function is ψ(r) = Ae-r/r1, where A is a certain constant, r1 is the first Bohr radius. 

Ans. We find A by normalization as above. We get

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Then the electronic charge density is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The potential Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE due to this charge density is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

so at the origin Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

 

Q.94. Particles of mass m and energy E move from the left to the potential barrier shown in Fig. 6.3. Find:
 (a) the reflection coefficient R of the barrier for E > U0;
 (b) the effective penetration depth of the particles into the region x > 0 for E < Uo, i.e. the distance from the barrier boundary to the point at which the probability of finding a particle decreases e-fold. 

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Ans. (a) We start from the Schrodinger equation Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

which we write as Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

and Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

It is convenient to look for solutions in the form

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

In region I (x < 0), the amplitude of Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE is written as unity by convention. In II we expect only a transmitted wave to the right, B = 0 then. So

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The boundary conditions follow from the continuity of Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

1 + R = A

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Then  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The reflection coefficient is the absolute square of R :

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(b) In this case Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE is unchanged in form but

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

we must have B = 0 since otherwise ψ(x) will become unbounded as Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEFinally

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Inside the barrier, the particle then has a probability density equal to

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

This decreases toIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE of its value in 

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

 

Q.95. Employing Eq. (6.2e), find the probability D of an electron with energy E tunnelling through a potential barrier of width l and height U0  provided the barrier is shaped as shown:
 (a) in Fig. 6.4;
 (b) in Fig. 6.5. 

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Ans. The formula is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Here V(x2) = V(x1) = E and V(x) >E in the region x2 > x > x1.

(a) For the problem, the integral is trivial

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

(b) We can without loss of generality take x = 0 at the point the potential begins to climb. Then

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Then Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

 

Q.96. Using Eq. (6.2e), find the probability D of a particle of mass m and energy E tunnelling through the potential barrier shown in Fig. 6.6, where U(x) = U0(1 — x2/l2).

Ans.  The potential is Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE  The turning points are

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Then  Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEEIrodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

The integral is

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Thus Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

Irodov Solutions: Wave Properties of Particles. Schrodinger Equation- 3 Notes | Study I. E. Irodov Solutions for Physics Class 11 & Class 12 - JEE

 

 

 

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