(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 radius . Would this proton behave as a wave or as a particle?
(a) The wavelength of a proton that has kinetic energy equal to its internal energy is
(b) The moving proton would behave like as a particle in nature.
The internal energy E of an object is
The equation for the kinetic energy ( KE) of an object traveling at relativistic velocities is
Where, m is rest mass, and c is the speed of light.
De Broglie's wavelength
Where, p is the momentum.
Relativistic effect in de Broglie's equation
Where, is the velocity at which the protons kinetic energy equals its internal energy, and is Lorentz factor.
Speed at which the internal energy become equal to the kinetic energy
Lorentz factor is given by a relation,
Substitute the value of velocity in wavelength ,
The wavelength of a proton that has kinetic energy equal to its internal energy is given by,
Substitute for c in the above equation to solve for
Hence, the wavelength of a proton that has kinetic energy equal to its internal energy is
The wavelength of a proton that has kinetic energy equal to its internal energy is
As the wavelength of the proton is smaller than the roughly size of the proton .
So, the moving proton would behave like as a particle in nature.
Consider the following function:
(a) Sketch this function. (Without loss of generality, assume that C is greater than B.) Calculate the Fourier transform .
(b) Show that for large is proportional to .
(c) In general, is not continuous. Under what condition will it be, and howdoes behave at large values of k if this condition holds?
(d) How does a discontinuity in a function affect the Fourier transform for large values of k?
Here we investigate the link between n and l, reflected in equation (7-33). (a) Show that if a classical point charge were held in a circular orbit about a fixed point charge by the Coulomb force, its kinetic energy would be given by (b) According to equation (7-30), the rotational kinetic energy in hydrogen is . Of course, r is not well defined for a “cloud”, but by using argue that the condition that l not exceed n is reasonable.
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’s constant. Imagine that a beam of particles of mass m and speed v, moving in the x direction, passes through a single slit of width w . Show that the angle at which the first diffraction minimum would be found ( , from physical optics) is proportional to the angle at which the particle would likely be deflected , and that the proportionality factor is a pure number, independent of m, v, w and h . (Assume small angles: ).
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