... CALC The WKB Approximation. It can be a challenge to solve the Schrödinger equation for the bound-state energy levels of an arbitrary potential well. An alternative approach that can yield good approximate results for the energy levels is the WKB approximation(named for the physicists Gregor Wentzel, Hendrik Kramers, and Léon Brillouin, who pioneered its application to quantum mechanics). The WKB approximation begins from three physical statements: (i) According to de Broglie, the magnitude of momentum pof a quantum-mechanical particle is . (ii) The magnitude of momentum is related to the kinetic energy Kby the relationship . (iii) If there are no nonconservative forces, then in Newtonian mechanics the energy Efor a particle is constant and equal at each point to the sum of the kinetic and potential energies at that point:

where xis the coordinate. (a) Combine these three relationships to show that the wavelength of the particle at a coordinate xcan be written as

Thus we envision a quantum-mechanical particle in a potential well U1x2 as being like a free particle, but with a wavelength

that is a function of position. (b) When the particle moves into a region of increasing potential energy, what happens to its wavelength? (c) At a point where E= U1x2, Newtonian mechanics says that the particle has zero kinetic energy and must be instantaneously at rest. Such a point is called a classical turning point,since this is where a Newtonian particle must stop its motion and reverse direction. As an example, an object oscillating in simple harmonic motion with amplitude Amoves back and forth between the points each of these is a classical turning point, since there the potential energy equals the total energy In the WKB expression for l1x2, what is the wavelength at a classical turning point? (d) For a particle in a box with length L, the walls of the box are classical turning points (seeFig. 40.8). Furthermore, the number of wavelengths that fit within the box must be a half-integer (see Fig. 40.10), so thatwhere n = 1, 2, 3,c. [Note that this is arestatement of Eq. (40.29).] The WKB scheme for finding the allowed bound-state energy levels of an arbitrary potential well isan extension of these observations. It demands that for an allowed energy E, there must be a half-integer number of wavelengthsbetween the classical turning points for that energy. Since the wavelength in the WKB approximation is not a constant but depends on x, the number of wavelengths between the classicalturning points a and b for a given value of the energy is the integralof 1>l1x2 between those points:

Using the expression for l1x2 you found in part (a), show that the WKB condition for an allowed bound-state energycan be written as

(e) As a check on the expression in part (d), apply it to a particle in a box with walls at x= 0 and x= L. Evaluate the integral

and show that the allowed energy levels according to the WKB approximation are the same as those given by Eq. (40.31). (Hint:

Since the walls of the box are infinitely high, the points x= 0 and x= Lare classical turning points for anyenergy E. Inside

the box, the potential energy is zero.) (f) For the finite square well shown in Fig. 40.13, show that the WKB expression given in

part (d) predicts the samebound-state energies as for an infinite square well of the same width. (Hint:Assume E6 U0 . Then the

classical turning points are at x= 0 and x= L.) This shows that the WKB approximation does a poor job when the potential-energy function changes discontinuously, as for a finite potential well. In the next two problems we consider situations in which the potential energy function changes gradually and the WKB approximation is much more useful.

Step by step solution

01

About de brogliewavelength

de Broglie wavelength is an important concept while studying quantum mechanics. The wavelength (λ) that is associated with an object in relation to its momentum and mass is known as de Broglie wavelength. A particle's de Broglie wavelength is usually inversely proportional to its force.

02

Determine the wavelength 

(a) First, we list the equations of the three physical statements of the WKB approximation:

Solving equation (i) for x\, equation (ii) for p and equation for K , we get:

Substituting equation (3) into equation (2), we get:

Substituting equation (3) into equation (2), We get:

Now, We substitute into equation (1), so we get:

therefore by solving the equation we get

03

Determine what happen to wavelength when the particle moves

(b) When the particle moves into a region of increasing potential energy, the quantity E decreases-

And in the WKB expression for A(m), A(m) is inversely proportional to the quantity

Therefore, as the particle moves into a region of increasing potential energy, the de Broglie wavelength increases-

04

Determine the wavelength A

(c) At a classical turning point,

Substituting into equation (4), we get:

therefore the wavelength A is 0

First, We reWrite the number of wavelengths between the classical turning points a and b for a given value of the ener

as given by the problem:

05

Determine the required expression

WNe sumply substitute equation (4) Into this Integral, so we get:

06

Determine the energy for the particle in the bpx

:e) For a particle in a box with Walls at a: : 0 and 1: = L, a = 0 and b : L; And sicne , the integral given by part

Solving for E, we get:

Which is the same as the energy given by equation 40-3]-

(f) For a particle in a potential well, in the region 0 < :1: < L, the potential energy is U(m) : 0;

This gives us the same result as in part (e)-

Therefore

(f) For a particle in a potential well in the region 0 < :1: < L, the potential energy is U(m) :

This gives the same result as in part (f)-

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