Ceballos Alex Dr.Cole 4/21/11
PHY 361 TEST #2
1. The book states in pg. 200 that the slope of the wavefunction must be continuous
wherever U(x) has a finite value. Since at x=0 and x=L the potential U(x) equals infinity, also known as a non-finite value, the discontinuity of the slope of the wavefunction does not violate the requirement imposed on the WF.
2. (This is addressing Q3) [p] is fuzzy not because it's changing in magnitude in time, but
because the sign flips from positive to negative as the particle bounces off the potential walls over and over again. By squaring the momentum operator, we get rid of the sign changing and hence obtain an eigenvalue for our eigenfunction. Thus [p2] is sharp.
3. (This is addressing Q4) No, the normalization coefficient remains the same. We must
remember that physically speaking the system is still the same as before, the overall length, or width of the box remains L. So when we impose the normalization condition, the second term of the integrand, the cos(2x) contributes nothing to the integral, so all you get is just 1, evaluated from [-L/2, L/2], which turns out to be the same as evaluating it in the interval [0,L] because of the symmetry of our system, and again, because the particle doesn't give a damn where we set our origin, the normalization coefficient is still the same.
4. (This is addressing Q5) The uncertainty principle determines that you can only know one
component of angular momentum, and the total magnitude, not the direction. The reason is that if you knew the direction of L, then the particle would be bound to an orbital plane out from which no momentum could exist, thus we would know a particle's linear momentum, in direct violation of the uncertainty principle. Not to mention that you have fixed its radius as well, a double violation of the uncertainty principle. Thus you can only know one component with absolute certainty, unless of course L=0, in which case all components are zero but this is more of a pedantic answer that lacks any physical interest. If you were to add the 3 vector components of L, one of these components would be sharp, but the other two are changing continuously as L precesses around a certain axis, so it's completely meaningless to try to get something out of this sum since two of its components are continuously changing in time.
5. (This is addressing Q6) The zero point energy is the ground state energy of a harmonic
oscillator and the interesting aspect of this concept is that despite its name, this energy is non-zero. The implication of this is its relationship with the uncertainty principle. If this zero-point energy were zero, then our particle would be dwelling at the bottom of a potential well, localized sharply with absolute certainty in momentum and position since our particle would not be moving, thus again violating the uncertainty principle.
6. (This is addressing Q7) A linear differential equation is one where each term has no
products or quotients of its dependent variable. In the context of the SE, having it be a linear differential equation allows one to obtain the discrete energy values quickly in a natural way, similar to the case of standing waves on a string. So Schrodinger's equation in a sense, retains some of its similarities with the wave of equation classical mechanics, not to mention that a solution of SE consists of a linear combination whose constants can be modified according to the needs of the system, in order to stitch together a wavefunction that is smooth, continuous and single valued. Thus it is extremely easy to apply to physical problems because it carries an internal consistency that is very similar to the classical wave equation.
7. (This is addressing Q8) In terms of energy Bohr got it right, both QM and Bohr state that
E=(-13.6 eV)/n2 for the hydrogen atom. For angular momentum however, QM's formulation of angular momentum can never be an integral multiple of Planck's constant as evidenced by a quick look to its formula L=sqrt(l(l+1))*h-bar, therefore Bohr got the angular momentum wrong. Lastly, Bohr essentially places the electron only on stable orbits that are multiples of his famous Bohr radius, so in his model, the electron can only be orbiting the nucleus in one of these quantized orbits. QM posseses a continuous distribution of the electron, weighted according to probability. Also QM states that the electron is not really orbiting the nucleus, so in terms of position, Bohr got it wrong as well.
8. (This is addressing Q10) Most probable distance is the value of r that maximizes the
probablity P(r), so if you were to probe the location of the electron, you'd probably find it at that value of r given by the calculation of the most probable distance. The average distance doesn't give you this information, the avg. distance is found by weighting each possible distance with the probability that the electron will be found at that same distance. So it more or less tells you how skewed the probability density curve is as measured from the most probable distance. Therefore if you have a non-symmetrical probability density, you'd get values for the average distance that are lesser or higher than the most probable distance depending on how the curve is shaped.
9. (This is addressing Q12) The amplitude of the WF increases at the turning points because those are the regions where the particle spends most of its time. We see in fig. 20 that as the energy increases, these turning points start to match up with the classical turning points of the harmonic oscillator because these are the spots where the particle has maximum potential energy. Thus this goes hand in hand with the correspondence principle as we see that with higher energies the probability is more concentrated on the classical turning points. So essentially, the probability of finding the particle in any element dx is proportional to how much time the particle spends in dx, thus the increase in amplitude is due to the particle's potential being maximal at those points, and these in turn correspond to where the particle spends most of its time, as clearly evident in higher quantum states.
10. (This is addressing Q13) The correspondence principle can be understood to be the
relationship between QM and classical physics, its connecting bridge more or less. Essentially it states that classical physics is the resutl of taking the limit of QM as n infinity. What we see in fig. 20 is a confirmation of this principle. As n increases, the quantum and classical probabilities start to agree with each other more and more.