That might seem surprising, but it lines up with what you expect for a classical mass on a spring- it's most likely to be found where it's moving slowly, near one of the extremes of the motion.Īs you go down in energy, though, you eventually hit a lowest possible energy state, the "ground state," which looks very different from anything you'd see classically- the most likely position is right at the center, with a substantial probability tailing off toward the outside. At high energy, these look more or less like you would expect- some wavy stuff happening in the central region (because quantum particles have to have wave nature), and the probability being highest toward the outside. The other weird thing to notice about these is the way the probability distributions behave as you go down in energy. You'll never observe a probability distribution that isn't one of these, and you'll never find an energy between these allowed levels. Each time you increase the energy by one unit, you change the probability distribution to the next plot up. This is why the figure for the quantum case doesn't have a single distribution, but eight different ones. The characteristic frequency of a quantum harmonic oscillator shows up through this relationship: there is a fundamental energy unit for any quantum harmonic oscillator, determined by the analogue of the mass and spring stiffness in the same way as the classical system, and you can only increase or decrease the energy by that unit. The "quantum" part of "quantum mechanics," though, refers to the fact that energy comes only in discrete amounts, and energy is related to frequency. This still has a characteristic resonance frequency, though it's not as immediately obvious as the nice sine-wave position graphs you can make for the classical mass on a spring. Probability distributions for the eight lowest energy states of a quantum harmonic oscillator. There is a characteristic frequency at which air in a confined space will naturally oscillate, and if you make a sound of the appropriate pitch, it will get louder and louder in the same way that flexing your knees at the right rate will set a playground platform bouncing. episode that kicked this off, is another example of a classical harmonic oscillator. If you shake them at the right frequency, they'll oscillate more dramatically, which is kind of fun if you're on a piece of playground equipment, but famously disastrous for a poorly designed bridge.Ī particular pitch resonating in a creepy underground chamber, as in the Agents of S.H.I.E.L.D. All of these share the same key behavior: they have a characteristic resonance frequency that depends on the parameters of the system in a straightforward way. And it turns out that you can use the same mathematics to describe a lot of other kinds of motion- a pendulum swinging back and forth, for example, provided the angle of the swing isn't too big, or a vibrating string in a musical instrument, or the flexing of support beams in some sort of structure. This is really "nice" behavior, in a mathematical sense, so the mass-on-a-spring system is much loved by physicists. The different oscillation frequencies reflect the large difference in mass. Position as a function of time for me on the platform (filled circles) and the unladen platform. The frequency of my bouncing is much lower than that of the platform alone, which is even more obvious if you do the position-versus-time thing again: You can see this most clearly in the second part of the video above, which shows me (well, my legs- I got this by just setting the camera on the ground) bouncing on the platform, then stepping off and letting it oscillate by itself.
This resonant frequency is a characteristic of the particular object doing the oscillation, and depends only on the mass of the object and the stiffness of the spring. There's a resonance behavior to this- he sets it into motion by flexing his knees, and if he wiggles at just the right rate, matching this characteristic frequency, the oscillation of the platform gets bigger and more fun. What's more, you can convince yourself, if you look closely, that there's a single characteristic frequency to this- he bounces up and down at the same rate all the time.
The steady decrease is because I'm not a great cameraman, and moved a bit while I was shooting, but the up-and-down bouncing is very clear. oscillation frequency, superimposed on a downward trend caused by the camera moving. The Pip's position as a function of time while bouncing.
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