2) quantum sphere on a spring / SurprizingFacts

The key result of the previous article was that the vibrational motion of a ball on a spring in the prequantum physics of Newton and his friends takes the form [$

$ z (t) = Z_0 + A cos [ 2 π ν t ] $ "data-tex =" display “/>

• z – position of the ball as a function of time t,
• z 0 – the position of the ball's equilibrium (where it would have rested if it had not wavered),
• A is the amplitude of the oscillations (which we can choose arbitrarily large or small),
• ν [ню] is the oscillation frequency (depending only on the force of the spring K and the mass of the ball M, and not depending on A).

In addition, the total energy stored in the oscillation is

 $ E = 2 π ^ 2 ν ^ 2 A ^ 2 M $ "data-tex =" display

By changing A, we can keep any amount of energy in vibration.

In quantum mechanics, everything changes. At first glance (and we do not need any more), only one thing changes – the statement that the amplitude of the oscillations can be chosen arbitrarily large or small. It turns out that this is not so. Accordingly, the energy stored in the oscillation can not be arbitrarily chosen.

Fig. 1

Quantization of the amplitude of oscillations

Max Planck, the famous physicist of the early 20th century, discovered that there is something quantum in the universe, and introduced a new constant of nature, which is called a constant plank, h. Every time you meet something in quantum mechanics, you will see h. Quantitatively,

 $ h = 6,626068  times 10 ^ {-34} m ^ 2 kg / s $ "data-tex =" display

]  – a very small value for ordinary human life. And now that comes out:

A quantum ball on a spring can fluctuate only with amplitudes

 $ A = (1/2 π)  sqrt {2 nh / ν M} $ " Data-tex = "display

Where n is an integer, for example, 0, 1, 2, 1798 or 2 348 979. Oscillations are not arbitrary, but quantized: we can call n the quantum of oscillations. Definition: we say that a sphere that oscillates with a quantum of n is in the n-th excited state. If the quantum is zero, we say that it is in the ground state.

To give you an idea of ​​what this means, the first five excited states, and the ground state, are shown (rather naive – you should not take the image too seriously) in Fig. 1. The minimum of possible oscillations occurs in the state n = 1. This is a vibration quantum; The fraction of a quantum does not exist. The ball can not oscillate less, unless it is in a state without hesitation, when n = 0.

Everything else, at first glance, is the same. But in fact the history of quantum mechanics is much more complicated! But while we can move away from this confusion and use almost 100% of the correct physics.

Why can not we say that fluctuations are quantized based on their experience? Because in everyday systems, quantization is too small. Take a real ball and a spring – say, the ball weighs 50 grams, and the frequency of its oscillations – once a second. Then the amplitude for one quantum, n = 1, will correspond to the amplitude

 $ A = (1/2 π)  sqrt {2 nh / ν M} = 1,8  times 10 ^ {- 16} m $ "data-tex =" display

This is a couple tens of a thousandth of a millionth of a millionth of a meter, or 10 times less than a proton! One quantum of oscillations will not move the sphere even over a distance of the order of the size of the atomic nucleus! No wonder that we do not see any quantization! If the ball moves to the visible distance, it contains a huge number of quanta – and for such large n, from our point of view, we can make any A, see Fig. 2. We can not measure A precisely enough to notice such subtle limitations on its magnitude.

Fig. 2. Amplitude of the oscillations A for the state n. For small n, the individual values ​​of A lie far apart, but already for n = 100 the allowed values ​​of A lie so close that the discreteness is already very difficult to observe. In everyday situations, the values ​​of n are so large that discreteness can not be noticed.

Note that in particular these values ​​are obtained due to the large mass of the ball. If the ball consisted of 100 iron atoms and would be a radius of one thousandth of a millionth of a meter, its minimum amplitude would be one millionth of a meter, that is, it would be a thousand times larger than its radius. And it's big enough to be seen in a microscope. But such a small ball would be exposed to forces working at atomic scales, and would oscillate much faster than once per second – and a large frequency corresponds to small amplitudes. So even with a small ball it is not easy to notice the quantization of nature.

Quantization of vibrational energy

Now take the quantization of the amplitude, and place it in the formula for the oscillation energy, which we already mentioned at the beginning of the article,

 $ E = 2 π ^ 2 ν ^ 2 A ^ 2 M $ "data-tex =" inline

. Substituting the allowed values ​​for A into it, we get an amazing result:

 $ E = n h ν $ "data-tex =" display

A surprisingly simple answer! The energy stored in the quantum sphere on the spring (naively speaking) is proportional to n, the number of vibration quanta, the constant bar h and the vibration frequency ν. More surprisingly, this simple formula is actually almost accurate! What does it show correctly?

• The energy that must be expended to increase the number of quanta in vibrations per unit, (n → n + 1), is h ν.
• In any oscillator encountered in everyday life, the energy of one quantum will be so small that we will never know about its quantization.

Let's check. For a ball with a spring oscillating once per second, one quantum of energy will be 6.6 × 10 -34 J, or 0.000 million 000 million 000 million 000 million 64,000 Joules. And Joule is the energy that you'll spend, picking an apple from the ground to the level of the belt – it's not that big! So this is an incredibly small amount of energy. Only in small molecules and even smaller systems can the frequency of oscillations be so large that the quantization of energy can be detected.

It turns out that the formula for energy is not entirely correct. Having performed real calculations for quantum mechanics, one can find that the correct formula for energy will be:

 $ E = (n + 1/2) h ν $ "data-tex =" display

We often do not need to pay attention to this small shift of n by 1/2. However, it is very interesting – it is with him that the entire intricacies of quantum mechanics begin. Is not that curious? Even if there are no oscillation quanta in the oscillator, when n = 0, it still contains a small amount of energy. It is called the energy of zero-point vibrations, or zero energy, and is taken from the basic jitter, the basic unpredictability, living in the very heart of quantum mechanics. Look at Fig. 3, which, inevitably schematically and inaccurately, tries to demonstrate how the jitter is responsible for zero energy. The ball moves randomly, even in the ground state. In the future, we return to zero energy, since it will lead us to the most profound problems of all physics.

Fig. 3. The fundamental unpredictability of quantum mechanics can be imagined as random jitter, which changes the position of the ball. He accidentally moves even in the ground state, and also affects the excited state, although with increasing n its influence is not so noticeable. The diagram is sketchy, and it should not be taken too seriously.

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