The new theories, if one looks apart from their mathematical setting, are built up from physical concepts which cannot be explained in terms of things previously known to the student, which cannot even be explained adequately in words at all. Like the fundamental concepts (e.g., proximity, identity) which every one must learn on his arrival into the world, the newer concepts of physics can be mastered only by long familiarity with their properties and uses.
P.A.M. Dirac (1930) Preface The Principles of Quantum Mechanics
We have always had a great deal of difficulty understanding the world view that quantum mechanics represents. At least I do, because I'm an old enough man that I haven't got to the point that this stuff is obvious to me. Okay, I still get nervous with it... You know how it is, every new idea, it takes a generation or two until it becomes obvious that there's no real problem... I cannot define the real problem, therefore I suspect there's no real problem, but I'm not sure there's no real problem.
R. P. Feynman as quoted in Genius (1992)
As the above quotations suggest, quantum mechanics is difficult, or perhaps impossible to understand (Steven Weinberg: "no one fully understands" [quantum mechanics]). Nevertheless, quantum mechanics is used on a daily basis by thousands of physicists, chemists and engineers. (E.g., Nobel prize winning chemist Linus Pauling: "There is no part of chemistry that does not depend, in its fundamental theory, upon quantum principles.") The ability to use quantum mechanics depends in part on mechanical mathematical skills ("doing the algebra", now made much easier with programs such as Mathematica) but more importantly on "physical intuition". Unfortunately our bedrock intuition -- based on classical mechanics -- is often at odds with quantum mechanics. This does not mean you must discard your hard-won classical intuition, rather you should treasure those parts that survive into quantum mechanics. These pages try find common ground between Newton's explanations and Schrödinger's explanations. In addition because of the revolution in computer algebra (e.g., Mathematica), these pages try to formulate problems in ways that allow easy translation into Mathematica code. Often simple practices (like the use of dimensionless variables) make understanding easier for both the computer and the human. Finally I should note that these pages are aimed at folks who want to do quantum mechanics. Like Dirac, I believe that the apprentice quantum mechanic gains facility by practice. I hope that you will do the examples, not just read the examples. A set of problems at the end of each "chapter" provide extentions of the examples. Feel free to write me if you have questions or comments!
In classical mechanics if we say that the particle has a position of 100±1, we mean that the particle has a position in the range: 99-101, we're just not sure where. In quantum mechanics if we say that the particle has a position of 100±1, we mean that the particle is simultaneously all over the range: 99-101. This ability to be "spread out" is not surprising for waves (in fact it would make little sense to say the wave is localized to a region smaller than a wavelength... a wave needs at least a wavelength to "wave"); here we apply this wave property to things like electrons, which are traditionally called "particles".
a=F/m=-g: acceleration is the result of applying the force; it can be calculated by the force divided by the particle's mass
z(t)=z0+v0t+½at2: the height of the object (z) depends on the initial height (z0), the initial velocity (v0) and time (t)
v(t)=v0+at: the velocity changes uniformly in time from its initial value
U(z)=-Fz=mgz: the potential energy has the property that if you take minus the derivative of it w.r.t. position, you get the force. -Fz + constant works; we've set the constant equal to zero.
In the case of the ball falling near the surface of the Earth, the above described motion cannot continue indefinitely as the ball soon encounters the ground. In the case of a perfectly elastic collision with the ground, the ground provides a force to exactly reverse the ball's velocity. It bounces forever between the ground (z=0) and some maximum height (zmax) that depends on its energy.
x(t)=A sin(t+): the particle oscillates around equilibrium getting as far away as ±A. The period, T (the time it takes to make one complete oscillation), is determined by (the angular frequency): T=2/. is in turn determined by the strength of the spring, k (called the spring constant) and the mass m: 2=k/m. Thus a strong spring connected to a light particle will oscillate quickly, i.e., with a short period.
v(t)=A cos(t+): the velocity (v) of the particle also oscillates, i.e., sometimes the particles is moving to the right (positive v) sometimes it is moving to the left (negative v). Notice that the particle has is maximum speed (of A) when the cosine term reaches its extremes of ±1. That happens only when sine is zero (because cos2+sin2=1) and hence the particle is moving through the equilibrium position (x=0). Similarly, the particle is momentarily at rest (v=0) only when the particle is at an extreme position (±A; i.e., if cosine is zero, sine must be ±1).
U(x)=½kx2: the potential energy has the property that if you take minus the derivative of it w.r.t. position, you get the force. ½kx2 + constant works; we've set the constant equal to zero.
Planets move in ellipses with the Sun at one focus. Ellipses can be described in terms of their semi-major axis, a, (basically the longest radius) and their eccentricity (basically how squashed the ellipse is: e=0 is a circle, a fully squashed ellipse looks like a line and has e=1).
r=a(1-e2)/(1+e cos()); rmin= a(1-e); rmax=a(1+e)
, the polar angle from closest approach is given the odd name: true anomaly. The timing of the motion (i.e., when the planet or electron has a particular ) is a bit complex. The game is to express the true anomaly () in terms of the eccentric anomaly (u) and then find an expression relating time and the eccentric anomaly. For nearly circular orbits it turns out that the true anomaly, the eccentric anomaly, and the mean anomaly (t) are all approximately equal to each other. I apologize for this archaic nomenclature, but physics is stuck with these names. Below find the geometric construction that relates the true anomaly and the eccentric anomaly, the formula relating these two, and the formula relating time and the eccentric anomaly.
In the above picture, the yellow dot represents the Sun (or a nucleus); it is at a focus (F) of the ellipse. The red parts of the diagram have to do with the geometric construction for the eccentric anomaly which is measured from the center (C) of the ellipse. There is nothing physically at the center of the ellipse, this is all just part of a geometric construction. The blue parts of the diagram relate the semi-major axis (a) and the distance between the ellipse center and the focus (ae). The semi-minor axis, b, can be related to the semi-major axis and the eccentricity:
b = a(1-e2)½
The black ellipse is the orbit of the particle, i.e., the set of positions the particle will traverse during a period (T=2/).
The velocity vector is, of course, changing as the position vector is changing. The set of velocities the particle will have during a period is called the hodograph; it's just like an orbit, but for velocity rather than position. The hodograph is surprising: it's just a circle, but the center of the circle is not v=0.
Note that at closest approach to the "Sun" (=0) the speed is a maximum and the velocity points in the y direction, whereas at the far point (=180°) the speed is a minimum and the velocity points in the -y direction. As the eccentricity approaches 1, the maximum speed gets arbitrarily large and the minimum speed approaches zero.
The above figures have been drawn with a rather large eccentricity: e=.707. For eccentricities similar to those of the planets it would be hard to distinguish the elliptical orbit from a circle (although for some planets--like Mars--the position of the Sun would look noticeably "off-center"...because the Sun is at a focus rather than the center).
If the (attractive) radial force is given by: Fr=-/r2, the potential energy is U(r)=-/r, so that the minus derivative of U is the force. Notice that just like the spring (which also attracts to the origin), the potential energy is ever smaller as you approach the origin. Unlike the spring, this potential energy is numerically negative (i.e., U(r)<0). That is a result of our choice of additive constant. For the spring we choose the potential to be zero at the origin, forcing the potential to be arbitrarily big as x approaches infinity. For this problem we take the potential energy to be zero as r approaches infinity, so the potential at the origin must be infinitely less, i.e., negative infinity.
Most introductory courses in quantum mechanics start out with problems that are not one of the above classical problems: "square well" potentials also known as particle-in-a-box problems, and delta function potentials.