Question: Explain to your friend. who has just learned about simple one-dimensional standing waves on a string fixed at its ends, why hydrogen's electron has only certain energies, and why, for some of those energies, the electron can still be in different states?
If the energy levels are limited, still waves can have different orientations and may result in a different state but of the same energy.
One-dimensional standing waves can be observed when a medium is having its opposite ends fixed, and nodes are located at the endpoints. The simplest example of the one-dimensional standing wave will be the one having only one antinode in the middle. This will be half of the wavelength.
The electron of hydrogen behaves as a wave of the bound state, that is why they have only certain energy or certain allowed standing waves. But when multiple dimensions are introduced in space, it is possible to have different standing waves but still have the same frequency/energy.
Assume a square wave of two dimensions for instance, the energy/frequency of the wave will be the same for both the following conditions,
(i) when the wave contains two waves along –axis and one wave along -axis
(ii) when the wave contains two waves along -axis and one wave along –axis.
Thus, if the energy levels are limited, still waves can have different orientations and may result in a different state but of the same energy.
Consider two particles that experience a mutual force but no external forces. The classical equation of motion for particle 1 is , and for particle 2 is , where the dot means a time derivative. Show that these are equivalent to , and . Where, .
In other words, the motion can be analyzed into two pieces the center of mass motion, at constant velocity and the relative motion, but in terms of a one-particle equation where that particle experiences the mutual force and has the “reduced mass” .
Classically, it was expected that an orbiting electron would emit radiation of the same frequency as its orbit frequency. We have often noted that classical behaviour is observed in the limit of large quantum numbers. Does it work in this case? (a) Show that the photon energy for the smallest possible energy jump at the “low-n-end” of the hydrogen energies is , while that for the smallest jump at the “high-n-end” is , where is hydrogen’s ground-state energy. (b) Use F=ma to show that the angular velocity of a classical point charge held in orbit about a fixed-point charge by the coulomb force is given by . (c) Given that , is this angular frequency equal to the minimum jump photon frequency at either end of hydrogen’s allowed energies?
Classically, an orbiting charged particle radiates electromagnetic energy, and for an electron in atomic dimensions, it would lead to collapse in considerably less than the wink of an eye.
(a) By equating the centripetal and Coulomb forces, show that for a classical charge -e of mass m held in a circular orbit by its attraction to a fixed charge +e, the following relationship holds
(b) Electromagnetism tells us that a charge whose acceleration is a radiates power . Show that this can also be expressed in terms of the orbit radius as . Then calculate the energy lost per orbit in terms of r by multiplying the power by the period and using the formula from part (a) to eliminate .
(c) In such a classical orbit, the total mechanical energy is half the potential energy, or . Calculate the change in energy per change in r : . From this and the energy lost per obit from part (b), determine the change in per orbit and evaluate it for a typical orbit radius of . Would the electron's radius change much in a single orbit?
(d) Argue that dividing by P and multiplying by dr gives the time required to change r by dr . Then, sum these times for all radii from to a final radius of 0. Evaluate your result for . (One limitation of this estimate is that the electron would eventually be moving relativistically).
94% of StudySmarter users get better grades.Sign up for free