1 - The fall of Classical Physicsdeangeli/fismod/qm.pdf · 2 3 Classical physics at the end of XIX...

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1 - The fall of Classical Physics

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Classical physics: Fundamental Models

n  Particle Model (particles, bodies) n  Motion in 3 dimension; for each time t, position and speed

are known (they are well-defined numbers, regardless we know them). Mass is known.

n  Systems and rigid objects n  Extension of particle model

n  Wave Model (light, sound, …) n  Generalization of the particle model: energy is transported,

which can be spread (de-localized) n  Interference

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Classical physics at the end of XIX Century n  Scientists are convinced that the particle and wave model can describe the

evolution of the Universe, when folded with n  Newton’s laws (dynamics) n  Description of forces

n  Maxwell’s equations n  Law of gravity. n  …

n  We live in a 3-d world, and motion happens in an absolute time. Time and space (distances) intervals are absolute.

n  The Universe is homogeneous and isotropical; time is homogeneous. n  Relativity

n  The physics entities can be described either in the particle or in the wave model.

n  Natura non facit saltus (the variables involved in the description are continuous).

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Something is wrong Relativity, continuity, wave/particle (I)

n  Maxwell equations are not relativistically covariant!

n  Moreover, a series of experiments seems to indicate that the speed of light is constant (Michelson-Morley, …)

A speed!

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5

Something is wrong Relativity, continuity, wave/particle (IIa)

n  In the beginning of the XX century, it was known that atoms were made of a heavy nucleus, with positive charge, and by light negative electrons n  Electrostatics like gravity:

planetary model n  All orbits allowed

n But: electrons, being accelerated, should radiate and eventually fall into the nucleus

s10

41

41

32

10

2

2

0

23

2

0

−≈⇒

⎟⎟⎠

⎞⎜⎜⎝

⎛==

−=⎟⎟⎠

⎞⎜⎜⎝

⎛=

τ

πε

πε

mre

mFa

dtdEa

ceW

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Something is wrong Relativity, continuity, wave/particle (IIb)

n  If atoms emit energy in the form of photons due to level transitions, and if color is a measure of energy, they should emit at all wavelengths – but they don’t

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Something is wrong Relativity, continuity, wave/particle (III)

n  Radiation has a particle-like behaviour, sometimes

n  Particles display a wave-like behaviour, sometimes

n  => In summary, something wrong involving the foundations: n  Relativity n  Continuity n  Wave/Particle duality

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Need for a new physics

n  A reformulation of physics was needed n  This is fascinating!!! Involved philosophy, logics, contacts

with civilizations far away from us… n  A charming story in the evolution of mankind

n  But… just a moment… I leaved up to now with classical physics, and nothing bad happened to me!

n  Because classical physics fails at very small scales, comparable with the atom’s dimensions, 10-10 m, or at speeds comparable with the speed of light, c ~ 3 108 m/s

Under usual conditions, classical physics makes a good job.

n  Warning: What follows is logically correct, although sometimes historically inappropriate.

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I Light behaves like a particle,

sometimes

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Photoelectric Effect Features and Photon Model explanation

n  The experimental results contradict all four classical predictions

n  Einstein interpretation: All electromagnetic radiation can be considered a stream of quanta, called photons

n  A photon of incident light gives all its energy hƒ to a single electron in the metal

⎟⎠

⎞⎜⎝

⎛ =≡=π

ω2hhfE

n  h is called the Planck constant, and plays a fundamental role in Quantum Physics

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The Compton Effect

n  Compton dealt with Einstein’s idea of photon momentum n  Einstein: a photon with energy E carries a

momentum of E/c = hƒ / c

n  According to the classical theory, electromagnetic waves of frequency ƒo incident on electrons should scatter, keeping the same frequency – they scatter the electron as well…

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n  Compton’s experiment showed that, at any given angle, a different frequency of radiation is observed n  The graphs show the scattered x-ray for

various angles

n  Again, treating the photon as a particle of energy hf explains the phenomenon. The shifted peak, λ‘> λ0, is caused by the scattering of free electrons

n  This is called the Compton shift equation

( )' 1 cosoe

hm c

λ λ θ− = −

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Compton Effect, Explanation

n  The results could be explained, again, by treating the photons as point-like particles having n  energy hƒ n  momentum hƒ / c

n  Assume the energy and momentum of the isolated system of the colliding photon-electron are conserved n  Adopted a particle model for a well-known wave

n  The unshifted wavelength, λo, is caused by x-rays scattered from the electrons that are tightly bound to the target atoms

n  The shifted peak, λ', is caused by x-rays scattered from free electrons in the target

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Particle-like behavior of light: now smoking guns…

n  The reaction

has been recorded millions of times…

e eγ + −→

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Summary n  The wave model cannot explain the behavior of light

in certain conditions n  Photoelectric effect n  Compton effect n  Blackbody radiation n  Gamma conversion/Bremsstrahlung

n  Light behaves like a particle, and has to be considered in some conditions as made by single particles (photons) each with energy

h ~ 6.6 10-34 Js is called the Planck’s constant E hf ω= ≡ h

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II Particles behave like waves,

sometimes

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Should, symmetrically, particles display radiation-like properties?

n  The key is a diffraction experiment: do particles show interference?

n  A small cloud of Ne atoms was cooled down to T~0. It was then released and fell with zero initial velocity onto a plate pierced with two parallel slits of width 2 µm, separated by a distance of d=6 µm. The plate was located H=3.5 cm below the center of the laser trap. The atoms were detected when they reached a screen located D=85 cm below the plane of the two slits. This screen registered the impacts of the atoms: each dot represents a single impact. The distance between two maxima, y, is 1mm.

n  The diffraction pattern is consistent with the diffraction of waves with

ph

=λ

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Diffraction of electrons

n  Davisson & Germer 1925: Electrons display diffraction patterns !!!

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de Broglie’s wavelength n  What is the wavelength associated to a particle?

de Broglie’s wavelength:

n  Explains quantitatively the diffraction by Davisson and Germer……

Note the symmetry

What is the wavelength of an electron moving at 107 m/s ?

(smaller than an atomic length; note the dependence on m)

h p kp

λ = ⇒ = h

( )( )( )

3411

31 7

6.63 10 Js7.28 10 m

9.11 10 kg 10 m/shmv

λ−

−

−

⋅= = = ⋅

⋅

kpωE

==

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Atomic spectra n  Why atoms emit according to a discrete energy spectrum?

2 2

1 1 1Per l'idrogeno interi

legata "numerologicamente" a h

H

H

R m nm n

Rλ

⎛ ⎞= − <⎜ ⎟⎝ ⎠ n  Something must

be there...

Balmer

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Electrons in atoms: a semiclassical model

n  Similar to waves on a cord, let’s imagine that the only possible stable waves are stationary…

2 π r = n λ n=1,2,3,…

2h nh

p pnr pr Lλ π= ⇒ = ⇒ == h

=> Angular momentum is quantized (Bohr postulated it…)

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2

2

2 2

2

2

22

2

ke k e

p e ek p

Emv e eF k E kr r r r

eE k E Er

eE kr

= = = ⇒ =

= = −− ⇒ + =

v

r m

F

NB:�  In SI, ke = (1/4πε0) ~ 9 x 109 SI units�  Total energy < 0 (bound state)�  <Ek> = -<Ep/2> (true in general for bound states, virial theorem)�

2 22 22

2 22 22 2

e

k en

e

L n mvrk em n

m e mr rE v kr

nr rk me

= =⎛ ⎞⇒ = ⇒⎜ = ≡⎟

= = ⎝ ⎠

hh

h

Only special values are possible for the radius !�

Hydrogen (Z=1)

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Energy levels

n  The radius can only assume values

n  The smallest radius (Bohr’s radius) is

n  Radius and energy are related:

n  And thus energy is quantized:

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2ne

r nk me

=h

2

2eeE kr

= −

22

2 20

1 13.6 eV2 2

en e

n

k eeE kr a n n

= − = − = −

2

1 02 .0529e

r nm ak me

= = ≡h

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Transitions n  An electron, passing from an orbit of energy Ei

to an orbit with Ef < Ei, emits energy [a photon such that f = (Ei-Ef)/h]

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Level transitions and energy quanta

0

2

2 21 1

2i f

ef i

E E ef kh a h n n

⎛ ⎞−= = −⎜ ⎟⎜ ⎟

⎝ ⎠

0

2

2 2 2 21 1 1 1 1

2e Hf i f i

f ek Rc a hc n n n nλ

⎛ ⎞ ⎛ ⎞= = − ≡ −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

n  We obtain Balmer’s relation!

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Limitations

n  Semiclassical models wave-particle duality can explain phenomena, but the thing is still insatisfactory, n  When do particles behave as particles, when do they behave

as waves? n  Why is the atom stable, contrary to Maxwell’s equations?

n  We need to rewrite the fundamental models, rebuilding the foundations of physics…

kpωE

==

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2a - The Schroedinger equation

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Wavefunction

n  Change the basic model! n  We can describe the position of a particle

through a wavefunction ψ(r,t). This can account for the concepts of wave and particle (extension and simplification).

n  Can we simply use the D’Alembert waves, real waves? No…

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Wavefunction - II

n  We want a new kind of “waves” which can account for particles, old waves, and obey to F=ma. n  And they should reproduce the characteristics of “real” particles: a

particle can display interference corresponding to a size of 10-7 m, but have a radius smaller than 10-10 m

n  Waves of what, then? No more of energy,

but of probability

n  The square of the wavefunction is the intensity, and it gives the

probability to find the particle in a given time in a given place.

n  Waves such that F=ma? We’ll see that they cannot be a function in R, but that C is the minimum space needed for the model.

dVtrdE 2),(Ψ∝

dVtrdP 2),(Ψ∝

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SUMMARY n  Close to the beginning of the XX century, people thought that

physics was understood. Two models (waves, particles). But: n  Quantization at atomic level became experimentally evident n  Particle-like behavior of radiation: radiation can be considered in some

conditions as a set of particles (photons) each with energy

n  Wave-like property of particles: particles behave in certain condistions as waves with wavenumber

n  Role of Planck’s constant, h ~ 6.6 10-34 Js n  Concepts of wave and particle need to be unified: wavefunction ψ (r,t).

E hf ω= ≡ h

/p h kλ= ≡ h (E, pc) = (ω,kc)

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L’equazione di Schroedinger

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Proprieta’ della funzione d’onda

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L’equazione di S.

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2b – The Schroedinger equation

  A particle of mass m on the x-axis is subject to a force F(x, t)

  The program of classical mechanics is to determine the position of the particle at any given time: x(t). Once we know that, we can figure out the velocity v = dx/dt, the momentum p = mv, the kinetic energy T = (1/2)mv2, or any other dynamical variable of interest.

  How to determine x(t) ? Newton's second law: F = ma. –  For conservative systems - the only kind that occur

at microscopic level - the force can be expressed as the derivative of a potential energy function, F = -dV/ dx, and Newton's law reads m d2x/dt2 = -dV/ dx

–  This, together with appropriate initial conditions (typically the position and velocity at t = 0), determines x(t).

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  A particle of mass m, moving along the x axis, is subject to a force F(x, t) = -dV/ dx   Quantum mechanics approaches this

same problem quite differently. In this case what we're looking for is the wave function, Ψ (x, t), of the particle, and we get it by solving the Schroedinger equation:

  In 3 dimensions,

~ 10-34 Js

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The statistical interpretation

  What is this "wave function", and what can it tell you? After all, a particle, by its nature, is localized at a point, whereas the wave function is spread out in space (it's a function of x, for any given time t). How can such an object be said to describe the state of a particle?

  Born's statistical interpretation:

Quite likely to find the particle near A, and relatively unlikely near B.

  The statistical interpretation introduces a kind of indeterminacy into quantum mechanics, for even if you know everything the theory has to tell you about the particle (its wave function), you cannot predict with certainty the outcome of a simple experiment to measure its position –  all quantum mechanics gives is statistical information

about the possible results   This indeterminacy has been profoundly disturbing

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Realism, ortodoxy, agnosticism - 1

  Suppose I measure the position of the particle, and I find C. Question: Where was the particle just before I made the measurement? There seem to be three plausible answers to this question…

1. The realist position: The particle was at C. This seems a sensible response, and it is the one Einstein advocated. However, if this is true QM is an incomplete theory, since the particle really was at C, and yet QM was unable to tell us so. The position of the particle was never indeterminate, but was merely unknown to the experimenter. Evidently Ψ is not the whole story: some additional information (a hidden variable) is needed to provide a complete description of the particle

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  Suppose I measure the position of the particle, and I find C. Question: Where was the particle just before I made the measurement?

2. The orthodox position: The particle wasn't really anywhere. It was the act of measurement that forced the particle to "take a stand“. Observations not only disturb what is to be measured, they produce it .... We compel the particle to assume a definite position. This view (the so-called Copenhagen interpretation) is associated with Bohr and his followers. Among physicists it has always been the most widely accepted position. Note, however, that if it is correct there is something very peculiar about the act of measurement - something that over half a century of debate has done precious little to illuminate.

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3. The agnostic position: Refuse to answer. This is not as silly as it sounds - after all, what sense can there be in making assertions about the status of a particle before a measurement, when the only way of knowing whether you were right is precisely to conduct a measurement, in which case what you get is no longer "before the measurement"? It is metaphysics to worry about something that cannot, by its nature, be tested. One should not think about the problem of whether something one cannot know anything about exists

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Realism, ortodoxy, or agnosticism?


  Until recently, all three positions had their partisans. But in 1964 John Bell demonstrated that it makes an observable difference if the particle had a precise (though unknown) position prior to the measurement. Bell's theorem made it an experimental question whether 1 or 2 is correct. The experiments have confirmed the orthodox interpretation: a particle does not have a precise position prior to measurement; it is the measurement that insists on one particular number, and in a sense creates the specific result, statistically guided by the wave function.

  Still some agnosticism is tolerated…

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Collapse of the wavefunction

  Suppose I measure the position of the particle, and I find C. Question: Where will be the particle immediately after?

  Of course in C. How does the orthodox interpretation explain that the second measurement is bound to give the value C? Evidently the first measurement radically alters the wave function, so that it is now sharply peaked about C. The wave function collapses upon measurement (but soon spreads out again, following the Schroedinger equation, so the second measurement must be made quickly). There are, then, two entirely distinct kinds of physical processes: "ordinary", in which Ψ evolves under the Schroedinger equation, and "measurements", in which Ψ suddenly collapses.

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Normalization

  |Ψ (x, t)|2 is the probability density for finding the particle at point x at time t. The integral of |Ψ (x, t)|2 over space must be 1 (the particle has to be somewhere).

  The wave function is supposed to be determined by the Schroedinger equation--we can't impose an extraneous condition on Ψ without checking that the two are consistent.

  Fortunately, the Schroedinger equation is linear: if Ψ is a solution, so too is A Ψ , where A is any (complex) constant. What we must do, then, is pick this undetermined multiplicative factor so that The integral of |Ψ (x, t)|2 over space must be 1 This process is called normalizing the wave function.

  Physically realizable states correspond to the "square-integrable" solutions to Schroedinger's equation.

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Will a normalized function stay as such?

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Expectation values

  For a particle in state Ψ, the expectation value of x is

  It does not mean that if you measure the position of one particle over and over again, this is the average of the results –  On the contrary, the first measurement (whose

outcome is indeterminate) will collapse the wave function to a spike at the value obtained, and the subsequent measurements (if they're performed quickly) will simply repeat that same result.

  Rather, <x> is the average of measurements performed on particles all in the state Ψ, which means that either you must find some way of returning the particle to its original state after each measurement, or you prepare an ensemble of particles, each in the same state Ψ, and measure the positions of all of them: <x> is the average of them.

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Momentum

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More on operators

  One could also simply observe that Schroedinger’s equations works as if

(exercise: apply on the plane wave). In 3 dimensions,

Compound operators

  Kinetic energy is

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Angular momentum

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3 – The time-independent Schroedinger equation

  Supponiamo che il potenziale U sia indipendente dal tempo

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ψ, soluzione della prima equazione (eq.agli autovalori detta equazione di S. stazionaria), e’ detta autofunzione

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1. They are stationary states. Although the wave function itself

does (obviously) depend on t, the probability density does not - the time dependence cancels out. The same thing happens in calculating the expectation value of any dynamical variable

3 comments on the stationary solutions: 1

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2. They are states of definite energy. In mechanics, the total energy is called the Hamiltonian: H(x, p) = mv2/2 + V(x). The corresponding Hamiltonian operator, obtained by the substitution p -> -i hbar ∂/∂x, is


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3. They are a basis. The general solution is a linear combination of separable solutions. The time-independent Schroedinger equation might yield an infinite collection of solutions, each with its associated value of the separation constant; thus there is a different wave function for each allowed energy. The S. equation is linear: a linear combination of solutions is itself a solution. It so happens that every solution to the (time-dependent) S. equation can be written as a linear combination of stationary solutions.

To really play the game, mow we must input some values for V


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The infinite square well

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Infinite square well, 2

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But…

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The uncertainty principle (qualitative)

  Imagine that you're holding one end of a long rope, and you generate a wave by shaking it up and down rhythmically.

  Where is that wave? Nowhere, precisely - spread out over 50 m or so.

  What is its wavelength? It looks like ~6 m

  By contrast, if you gave the rope a sudden jerk you'd get a relatively narrow bump traveling down the line. This time the first question (Where precisely is the wave?) is a sensible one, and the second (What is its wavelength?) seems difficult - it isn't even vaguely periodic, so how can you assign a wavelength to it?

x(m)

x(m)

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  The more precise a wave's x is, the less precise is λ , and vice versa. A theorem in Fourier analysis makes this rigorous…

  This applies to any wave, and in particular to the QM wave function. λ is related to p by the de Broglie formula

  Thus a spread in λ corresponds to a spread in p, and our observation says that the more precisely determined a particle's position is, the less precisely is p

  This is Heisenberg's famous uncertainty principle. (we'll prove it later, but I want to anticipate it now)

The uncertainty principle (qualitative, II)

x(m)

x(m)

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Exercise   A particle is represented at t=0 by the wavefunction

Ψ (x, 0) = A(a2-x2) |x| < a (a>0). = 0 elsewhere

a Determine the normalization constant A b, c What is the expectation value for x and for p at t=0?

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Exercise (cont.)   A particle is represented at t=0 by the wavefunction

Ψ (x, 0) = A(a2-x2) |x| < a (a>0). = 0 elsewhere

d, e Compute <x2>, <p2> f, g Compute the uncertainty on x, p h Verify the uncertainty principle in this case

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Free particle (V=0, everywhere)

31 (however, for any finite volume V, however large, ψ is normalizable)

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Let us assume that φ(k) is narrowly peaked about some particular value k0. Since the integrand is negligible except in the vicinity of k0, we may Taylor-expand the function ω(k) about that point and keep only the leading terms:

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Gradino di potenziale

a.  E < U0 b.  E > U0

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a. E < U0

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b. E > U0

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Barriera di potenziale – Effetto tunnel

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Stati legati a L=0

When we solve the Schroedinger equation in 3 dimensions for a central potential (for example, the Coulomb potential: electrons in an atom), we find among the solutions eigenstates of the angular momentum with L = 0. Thus, in QM, non-rotating bound states are possible! This explains the stability of the atoms (electrons In the ground states are bound without the need of a centripetal acceleration, and thus do not radiate). The ground state of the hydrogen atom has the smallest radius compatible with the uncertainty principle: EK =

p2

2m=

14πεo

e2

2r=> Δp ~ 2m 1

4πεoe2

2rΔx ~ r

we obtain, for r ~ 0.05 nm:

ΔpΔx ~ mre2

πεo~

1 - The fall of Classical Physicsdeangeli/fismod/qm.pdf · 2 3 Classical physics at the end of XIX...

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