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أحاديث في ميكانيك الكوانتم/3 — Transcript

by Basil Altaie · 2,520 words · 368 segments · language en · Watch on YouTube

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0:09In the first half of the twentieth
0:11century. Specifically in the first
0:16quarter, it became clear to physicists
0:18that particles have wave-like
0:19properties. At the same time, waves
0:27have particle-like properties. As we
0:31showed in the previous first and second
0:34lectures, light is a wave by nature.
0:37Yet it possesses particle-like
0:39properties. This was revealed by
0:42Planck’s experiments, the
0:44photoelectric effect, and Compton
0:46scattering. At the same time, electrons
0:50are particles, yet they possess
0:52wave-like properties. We stated that
0:57the one who connected the wave and the
0:59particle in a formal way is the
1:01Frenchman de Broglie. Where he
1:04established the relationship lambda
1:06equals h over p. But this is not enough
1:08. It is required to produce a
1:13comprehensive theory that explains all
1:15these phenomena, as we said. And one
1:20based on clear theoretical foundations.
1:23I mean, these phenomena are still just
1:25phenomena. They are still precursors to
1:29a theory that will come later. In fact,
1:38here we stand before the properties; we
1:40should reflect on the wave properties
1:42and particle properties to see if we
1:44need to make the particle a wave or the
1:46wave a particle. Here we need to
1:56reflect on some properties related to
1:58the wave and the particle. A wave
2:10manifests properties including
2:12reflection, refraction, interference,
2:14and polarization, whereas particles are
2:16characterized by having momentum,
2:18transferring energy through their
2:20movement in space, and being able to
2:22exchange energy with each other when
2:24they collide; but they do not interfere
2:27, for example, and particles do not
2:29usually show wave properties. But since
2:43particles in the microscopic world,
2:45such as electrons, neutrons, and
2:47protons, have shown wave-like
2:49properties. This invites us to derive.
2:59A mechanics that treats the particle as
3:01a wave. Well, the question before us
3:06now is: What is the main characteristic
3:08of a wave? And what is the main
3:12characteristic of a particle? We can
3:16say that a wave is characterized by
3:18being. I mean, first, extended; second,
3:25it is in a state of constant change,
3:27while the particle is characterized by
3:30the opposite: that it is localized,
3:32fixed, and stable. So, the wave is
3:40characterized by extension and constant
3:43change, and the particle is
3:45characterized by localization, meaning
3:47it occupies a limited, known space, and
3:50is mostly fixed and static—change is
3:52slight and slow. Well, if we now want
4:00to give the wave particle-like
4:02properties, the first property we must
4:04address is localization. Change is not
4:12a problem because the particle also,
4:14when it moves from one place to another
4:17or when conditions change, changes; it
4:19can be a function of time, meaning its
4:21state changes. However, localization?
4:29Is a primary characteristic. Therefore,
4:35if we want to create a wave description
4:37for a particle, we must localize the
4:39wave. How is a wave localized? This is
4:47the question. How is a wave localized?
4:51To answer this question, we recall that
4:53there are types of waves. I mean, waves
4:58are not just this form that you see,
5:01sine or cosine, the sinusoidal wave. A
5:06wave can be in any shape. Any
5:08disturbance. By definition, a wave is a
5:11disturbance. We could have a wave that
5:16is extended widely from infinity to
5:18negative infinity. And we could have a
5:24wave that is confined, confined between
5:26two points, or two walls. And we call
5:30this what? A standing wave, also called
5:33a stationary wave. Another type that we
5:37can also call waves is the pulse. A
5:45pulse can take on different shapes. Any
5:49shape at all, including geometric forms
5:52like square or sawtooth, etc. Any shape
5:55whatsoever; these are all waveforms.
6:02Therefore, if we want to ask again, how
6:04do we bias a wave? If we look at or
6:10contemplate any waveform, let's say the
6:13biased one that comes as a pulse,
6:18coming in the form of a pulse. This
6:25pulse, whatever its shape, is in
6:27reality a very large, possibly infinite
6:29, collection of standard waves.
6:37Standard waves mean trigonometric forms
6:40like sine and cosine. This is what
6:45Fourier analysis shows us. Fourier
6:52analysis proved that any shape can be
6:54represented as a huge collection of
6:56periodic waves. Right, taking their
7:03mathematical form as trigonometric, as
7:06we said, cosine or sine waves. Thus, if
7:14we take a wave y1 equal to amplitude A
7:17times cosine (kx-ωt). And we take
7:28another wave y2 that differs from the
7:30first by a small difference in
7:32wavelength and frequency. For example,
7:45let y2 have the same amplitude, but be
7:48cosine ((k + Δk) x-(ω + Δω) t). If
7:59we add these two waves, what will
8:01appear to us? What appears before us
8:10now is that their sum is a third wave,
8:12which is a disturbance; if we examine
8:14it closely, we find two sets of waves.
8:28A set with high frequency and short
8:30wavelength called phase waves, and
8:31another set of waves representing the
8:33envelope that wraps around these phase
8:36waves, which we call the group wave.
8:47This group wave is exactly what de
8:49Broglie meant by expressing a particle
8:52as a wave. And this de Broglie lambda,
8:59λ = h/p, is the wavelength λ of the
9:01group wave, not the phase wave. Alright
9:07, this is that. On the other hand, we
9:11see that in reality, and we can derive
9:14this, the group wave velocity is not
9:16the same as...phase, as the group
9:23velocity, which represents the particle
9:26, the wave that represents the particle
9:28, is the speed of the particle itself,
9:30v. Whereas the speed of phase waves, if
9:40you derive it, is c²/v. We might be
9:47surprised here to see that the phase
9:49wave speed has become greater than the
9:51speed of light. Therefore, and because
9:57of this, this result has implications.
10:02We will talk about them in later
10:03lectures, God willing. The important
10:07thing. If we add two waves, as shown,
10:12we create a wave packet. We call it, I
10:17mean, a capsule of waves. Okay, if we
10:20add two, three, four, five, six, 10,
10:23and a million waves. The more waves we
10:28add, the more compressed the wave
10:30packet becomes. Even more. Meaning the
10:37width becomes smaller. And so, if we
10:42said we want to add an infinite number
10:44of waves. We mean plane waves by that.
10:52The plane wave is represented by
10:55amplitude A multiplied by e ^ (i (kx-
10:58ωt)), for example. Okay. Now, if we
11:05only take a fixed time. Meaning, we
11:08remove the time factor and take it
11:10outside. We take only the spatial part,
11:15which is e to the i k x. And we sum an
11:20infinite number of these waves that
11:22differ by one d k from each other. What
11:27are we summing? We sum by performing an
11:30integral; over whom do we perform the
11:32integral? Consequently. And d,
11:36consequently, i x. d. From minus
11:44infinity to infinity. It is as if we
11:48are summing an infinite number of waves
11:50. The difference between one and the
11:53other is d k. This is the formula in
11:56front of you. And what does this give
11:58us? It gives us the delta function, the
12:02Dirac delta function, if you notice.
12:07Meaning it gives us a vertical line on
12:09the x-axis. This vertical line is the
12:16Dirac delta function at x equals 0. So,
12:19if we want to represent a point
12:21particle, we must sum an infinite
12:23number of waves for this point particle
12:25. However, if the particle has a size,
12:32dimensions, or boundaries. Then it is
12:36sufficient to sum a limited number of
12:37waves that interfere with each other.
12:40So, notice the particle now has a wave
12:43representation. It is what? In this
12:47concept, the particle has become a
12:49large sum of waves. A large group of
12:54waves interfering with each other to
12:56give us this entity. Which usually
13:00takes the form of a Gaussian shape and
13:02has a width. This width, in reality,
13:08represents the uncertainty in
13:10determining the particle's position.
13:15This is delta x, which is the width of
13:18the Gaussian pulse. As the figure shows
13:22, it represents delta x and. And of
13:28course, this delta x is what, if we
13:30later perform a Fourier transformation
13:32on it, gives us delta k. And delta k
13:39times delta x always equals, within the
13:42limits of unity. Order of 1, This is
13:49the idea, I mean, it equals
13:51approximately one; it is the idea. Or
13:55the idea from which the uncertainty
13:57principle originated, therefore.
14:02Therefore, If we represent a particle
14:08as a wave, we sacrifice an important
14:10and serious issue. Which is that we
14:17will no longer be able to determine the
14:19position and momentum of a particle
14:21simultaneously with infinite precision.
14:29We must sacrifice this, and thus
14:31uncertainty arises. Notice, therefore,
14:35where the Heisenberg Uncertainty
14:37Principle originated. This is exactly
14:40it, called the Heisenberg Principle, or
14:42the principle of uncertainty. Delta x
14:49times delta k is greater than or equal
14:52to 1. Now, if we remember that k is 2
14:54pi over lambda. And if we remember that
15:00lambda is h over p. Then it is clear
15:07that delta x times delta p will be of
15:09the order of h-bar. In fact, it is of h
15:15here; in reality, it is of the order of
15:18h-bar. In reality, it comes out greater
15:21than or equal to h-bar over 2 in
15:23detailed, precise derivations. I am
15:26providing a simple derivation here.
15:29Meaning an introduction to the subject,
15:32because as I said, these lectures are
15:34intended to introduce concepts. And the
15:38purpose is not to delve into detailed
15:40derivations. So, we now have a clear
15:47representation for the; it is possible
15:49to represent the particle. By a wave
15:56function, let's say we call it psi now,
15:58of x and t. Equals an amplitude A.
16:05Multiplied by e to the i k x minus
16:07omega t. A, e, parenthesis x minus
16:12omega t. This is the wave function, as
16:16you can see. Well, here, and in the
16:20year 1924 The famous Austrian physicist
16:29Erwin Schrödinger. Who said, let us
16:36look at this formula. Psi of x and t
16:40equals A times e to the i (kx minus
16:43omega t). If we know that omega is, in
16:49fact, E over h-bar. And we know that k
16:57is p over h-bar. Then we can write Psi
17:03of x and t. Equal to the amplitude
17:08multiplied by e to the i (px minus Et)
17:15all divided by h-bar. Let us now take
17:22the partial derivative with respect to
17:24time for Psi of x and t. What do we get
17:28? What we get, as is clear before you,
17:33Is that d-Psi by dt will equal minus iE
17:36over h-bar Times Psi of x and t. Now,
17:43what if we take the derivative with
17:45respect to x, for position? We will get
17:51partial Psi by partial x. Which will
17:56equal i times p Over h-bar times Psi of
18:01x and t. And if we repeat that, we
18:05would get the second partial With
18:12respect to x squared. It will equal
18:14minus p squared over h-bar squared
18:17times Psi of x and t. Now, what did
18:21Schrödinger do? He said, let us form a
18:27wave equation from these derivatives.
18:33The wave equation. How do we derive it?
18:38We derive it from the principle of
18:39total energy. Total energy equals
18:44kinetic energy plus potential energy. E
18:53equals p squared over 2m plus V. Which
18:55is the potential energy. Total energy
19:02equals kinetic plus potential. So, what
19:05happens? We have here p squared over 2m
19:08. It will become minus h-bar squared
19:12over 2m times the second partial of Psi
19:14with respect to x. We keep V as it is.
19:22And the right side of the equation,
19:25which is E-Psi, will become i*h-bar
19:27times the partial of Psi with respect
19:29to t. And this completes our equation.
19:36Which reads now minus h-bar squared
19:39over 2m Times the second partial of Psi
19:46with respect to x. Plus V times Psi
19:50equals E times Psi, Which is i*h-bar
19:52times the partial of Psi with respect
19:55to time. This is called the
20:02time-dependent Schrödinger equation.
20:06Where Psi is Psi of x and t. But if Psi
20:14, if it is possible to write Psi of x
20:16and t as Psi of x multiplied by e to
20:19the minus i Et over h-bar. Then we can
20:29eliminate time from both sides of the
20:31equation. And we get what is called the
20:36time-independent Schrödinger equation.
20:39Which is the equation that applies to
20:41Psi of x only. The wave function Psi of
20:44x. So it becomes minus h-bar squared
20:49over 2m times d2Psi of x over dx
20:52squared. Plus V times Psi of x equals E
20:58times Psi of x. And this is the
21:05time-independent Schrödinger equation.
21:12Of course, this equation is used to
21:14solve certain problems. In which the
21:19time dependence is separable from the
21:21spatial dependence. And we physically
21:25have a stationary state. As we will
21:29show in other lectures, God willing.
21:35The important thing is that this is the
21:37time-dependent Schrödinger equation.
21:43Of the notation. If V equals 0, it
21:47represents the free particle. Meaning
21:53the free particle is represented by
21:56minus h-bar squared over 2m, partial
21:59with respect to x twice, of psi of x, t
22:02equals i h-bar, partial with respect to
22:04t of psi of x, t. This is the free
22:12particle equation. And if the particle,
22:17or the system, is not dependent on time
22:19. Then, minus h-bar squared over 2m, d
22:26squared psi by dx squared, equals E psi
22:30. Equals E times psi of x. This is in
22:38the case of The components and systems
22:43that are not time-dependent. The
22:48important thing now is to understand
22:49the Schrödinger equation. What does
22:53the Schrödinger equation mean, simply?
22:57As we said, it originates fundamentally
23:00from the law of conservation of
23:01mechanical energy. Total energy equals
23:06kinetic plus potential energy. That is
23:10all there is to it. Here is the
23:11equation. So what does it represent or
23:15describe? It describes the motion of a
23:19particle. Or it describes a particle
23:23that has energy E. And, I mean. It
23:30moves with momentum p. And it moves
23:37within a field V, under the influence
23:39of a force or a force field V. Because
23:41V is the potential energy. And
23:47potential energy comes from the force
23:48field. Meaning, in the presence of a
23:51force field. If there is no force field
23:54, there is no V, which we call
23:56potential energy. This is what the
24:02Schrödinger equation represents, okay.
24:06What do we benefit from this equation?
24:09What is the use of the equation? The
24:10benefit is very great. If we can
24:15identify V, the potential in which the
24:17particle is located. Then we can solve
24:24The differential equation. It is a
24:26second-order differential equation with
24:28respect to space. And of the first
24:30order with respect to time. If we solve
24:32it, we can solve for psi. We obtain psi
24:34of x and t. And then what? Then what if
24:38we obtain psi? From this psi of x, t,
24:43we can obtain a lot of information. And
24:48this is the subject of our next lecture
24:50, God willing.

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