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Chapter 1: Q50E (page 1)

If things really do have a dual wave-particle nature, then if the wave spreads, the probability of finding the particle should spread proportionally, independent of the degree of spreading, mass, speed, and even Planck鈥檚 constant. Imagine that a beam of particles of mass m and speedv, moving in the x direction, passes through a single slit of width w . Show that the angle 胃1at which the first diffraction minimum would be found ( n位=wsin胃n, from physical optics) is proportional to the angle at which the particle would likely be deflected 胃鈮�鈭�pyp , and that the proportionality factor is a pure number, independent of m, v, w and h . (Assume small angles: sin胃鈮�tan胃鈮�胃).

Short Answer

Expert verified

It is shown that胃1=4蟺胃鈬�胃1伪胃

Step by step solution

01

The diffraction minimum:

It is known that the diffraction minimum can be obtained at胃1=位w

02

Required Proof:

Since, 胃1is a very small angle, then

胃1鈮�胃1

Here,胃1is the angle at which the first diffraction minimum is obtained. So,

胃1=位w胃1=hmvw

Now, the angle at which the particle deflected is

胃=鈮�鈭�pyp 鈥�.. (1)

Since, it is known that

鈭�y鈭�py鈮�h2

And the momentum, p = mv

So, substitute the above value into the angle equation (1), and you have

胃鈮�h(2鈭�y)(mv)鈮�h2wmv鈮�h4蟺尘惫飞

Now, from the above calculations, you can say that 胃1=4蟺胃鈬�胃1伪胃and the proportionality factor is a pure number equals 4蟺.

Hence, it is proved that 胃1伪胃and the proportionality factor is a pure number.

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Most popular questions from this chapter

It is shown in section 6.1 that for the E<U0 potential step, B=-伪+ik伪-ikA. Use it to calculate the probability density to the left of the step:

|蠄x<0|2=|Aeikx+Be-ikx|2

  1. Show that the result is, 4|A|2sin2(kx-胃)where 胃=tan-1(k/伪). Because the reflected wave is of the same amplitude as the incident, this is a typical standing wave pattern varying between 0and 4A*A.
  2. Determine data-custom-editor="chemistry" 胃and Din the limits kanddata-custom-editor="chemistry" 伪tend to 0and interpret your results.

Make a rough sketches of the wave functions at the top of the n=2 band and the bottom of the n=3 band for a one dimensional 鈥渃rystal鈥� consisting of seven finite wells. Explain why these two states of roughly equal wave number have vastly different energies.

Just what is stationary in a stationary state? The particle? Something else?

With reckless disregard for safety and the law, you set your high-performance rocket cycle on course to streak through an intersection at top speed . Approaching the intersection, you observe green (540 nm) light from the traffic signal. After passing through, you look back to observe red (650 nm) light. Actually, the traffic signal never changed color-it didn't have time! What is the top speed of your rocket cycle, and what was the color of the traffic signal (according to an appalled bystander)?

Exercise 81 obtained formulas for hydrogen like atoms in which the nucleus is not assumed infinite, as in the chapter, but is of mass,m1 whilem2is the mass of the orbiting negative charge. (a) What percentage error is introduced in the hydrogen ground-state energy by assuming that the proton is of infinite mass? (b) Deuterium is a form of hydrogen in which a neutron joins the proton in the nucleus, making the nucleus twice as massive. Taking nuclear mass into account, by what percent do the ground-state energies of hydrogen and deuterium differ?
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