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Chapter 1: Q86E (page 1)

Prove that fur any sine function sin(kx+蠒)of wavelength shorter than 2a, where ais the atomic spacing. there is a sine function with a wavelength longer than 2a that has the same values at the points x = a , 2a , 3a . and so on. (Note: It is probably easier to work with wave number than with wavelength. We sick to show that for every wave number greater than there is an equivalent less than 蟺/a.)

Short Answer

Expert verified

It is proved that there is a sine function with a wavelength less than 2a.

Step by step solution

01

Sum of angle and expression for θ:

The atomic spacing is a.

The expression of sum of angles is given by,

胃1+胃2=n蟺

The expression for is given by,

胃=kx+蠒

02

Determine sum of angles:

The expression of sum of angles is calculated as,

胃1+胃2=n蟺nk1x+蠒+k2x+蠒=n蟺nk1x+k2x+2蠒=n蟺n..........(1)

For

蟺ana+蠒=n蟺2蠒=n蟺2-n蟺=-n蟺2

03

solve further:

Substitute -n蟺2for 蠒in equation (1).

k1+k2x+2-n蟺2=n蟺k1+k2na=2苍蟺k1+k2=2蟺ak1=2蟺a-k2

Now since,

k1>蟺a,2蟺a-k2>蟺a,k2<蟺a

Therefore, it is proved that there is sine function with wavelength less than 2a.

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Most popular questions from this chapter

Question: Consider an electron in the ground state of a hydrogen atom. (a) Calculate the expectation value of its potential energy. (b) What is the expectation value of its kinetic energy? (Hint: What is the expectation value of the total energy?)

A particle is subject to a potential energy that has an essentially infinitely high wall at the origin, like the infinite well, but for positive values of x is of the form U(x)= -b/ x, where b is a constant

(a) Sketch this potential energy.

(b) How much energy could a classical particle have and still be bound by such a potential energy?

(c) Add to your sketch a plot of E for a bound particle and indicate the outer classical tuning point (the inner being the origin).

(d) Assuming that a quantum-mechanical description is in order, sketch a plausible ground-state wave function, making sure that your function's second derivative is of the proper sign when U(x)is less than E and when it is greater.

Question: Is the potential energy of an electron in a hydrogen atom well defined? Is the kinetic energy well defined? Justify your answers. (You need not actually calculate uncertainties.)

(a) Find the wavelength of a proton whose kinetic energy is equal 10 its integral energy.

(b) ' The proton is usually regarded as being roughly of radius10-15m. Would this proton behave as a wave or as a particle?

Nuclei of the same mass number but different Zare known as isobars. Oxygen-15 and nitrogen- 15 are isobars.

(a) In which of the factors considered in nuclear binding (represented by terms in the semi empirical binding energy formula) do these two isobars differ?

(b) Which of the isobars should be more tightly bound?

(c) k your conclusion in part (b) supported by the decay mode information of Appendix 1? Explain.

(d) Calculate the binding energies of oxygen-15 and nitrogen-15. By how much do they differ?

(e) Repeat part (d) but use the semi empirical binding energy formula rather than the known atomic masses.
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