Chapter XXI Quantum Mechan

A particle shoud have a wavelength related to its momentum in exactly p = the de Broglie wavelength And the relation between frequency and particle’s energy is also in the same way a

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GENERAL PHYSICS III

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Chapter XXI

§ 1 The wave nature of particles

§ 2 The Heisenberg Uncertainty Principle

§ 3 The Schrödinger equation

§ 4 Solutions for some quantum systemsNitro PDF Trialwww.nitropdf.com

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 It has been known from the previous chapter that light, and in general,electromagnetic waves have particle behavior.

 Some latter time than the quantum theory of light, it was discoveredthat particles show also wavelike behavior

The wave-particle duality of matter is the fundamental concept ofmodern physics

Newton’s classical physics should be replaced by the new mechanicswhich is able to describe the wave nature of particles

“QUANTUM MECHANICS”Nitro PDF Trialwww.nitropdf.com

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§1 The wave nature of particles:

1.1 De Broglie hypothesis:

In 1923, de Broglie put a simple, but extremely important idea which

initiated the development of the quantum theory He proposed that,

if light is dualistic (behaving in some situations like waves and in otherslike particles) → this duality should also hold for matter It means thatelectrons, alpha particles, protons,…, which we usually think of as

particles, may in some situations behave like waves

A particle shoud have a wavelength related to its momentum in exactly

p

= (the de Broglie wavelength)

And the relation between frequency and particle’s energy is also in

the same way as photon

= E

h

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1.2 Macroscopic and microscopic world:

Question: Why do we not observe the wave nature of particles in ourexperience of the macroscopic world?

The answer is given by two following examples:

Apply the de Broglie hypothesis to two cases, the first on the

macroscopic scale, and the second on the microscopic scale

The case of macroscopic particles:

A particle with m = 10 kg, v = 10 m/s  p = 100 kg.m/s

 = h/p = (6.63x10-34 / 100) m = 6.63x10-36 m

With this scale of wavelengths a mcroscopic particle can not

produce any observable effect of interference or diffraction

The case of microscopic particles:

An electron with m = 9.1x10-31 kg, accelerated to v = 4.4x106 m/s

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1.3 Electron diffraction:

 Davisson & Germer

experiment (1927):

• Elecrons emitted thermally

from the cathode C

• Then they are accelated by

a voltage V  a parallel beam

of monoenergetic electrons

are produced

• A plate P & a diagraph D plays the role of

a detector which measures the number ofscattered electrons

The experimental graph shows the angulardistribution of the number of scatteredelectrons (for V = 54 volts)

There is a peak at = 50o

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 Explanation:

• The existence of the peak at = 50o proves qualitatively &

quantitatively the de Broglie hypothesis !

Such a peak can only explained as a constructive interference of wavesscattered by the periodically placed atoms

• With electron beam of such low intensity that the electron go throughthe apparatus one at a time the interference pattern remains the same

 the interference is between waves associated with single electron

• For a quantitative consideration, we calculate the electron wavelength:

by using = h/p, where eV = p2/2m  p = 2meV

Substuting V = 54 volts, one gets = 1.67x10-10 m

by using the formula for the first order diffraction peak

= d sin where d was detemined from X-rays diffraction

experiments, d = 2.15x10-10 m

For = 50o, one gets = 1.65x10-10 m

The two obtained values of agree within the accuracy with experiment !

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1.4 Application of Matter Waves: Electron Microscopy

 The ability to “resolve” tiny objects improves as the wavelength

decreases Consider the microscope objective:

A good microscope objective has f/D 2, so with ~ 500 nm

the optical microscope has a resolution of dmin 1 m

D

f f

dmin  c  1 22 

• Nominal (conventional) minimumangle for resolution:

• The minimum d for which we can still resolve two objects is

c times the focal length:

Dc

= focal length of lens

We can do much better with matter waves because, as electrons withenergies of a few keV have wavelengths less than 1 nm

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 Example: Observation of a virus by an electron microscopy

 nm . k eV

nm eV

m

h

01640

5051

2 2

dmin  1 22 

nm f

D nm

f

D

d 0 0164

22 1

2 22

1

To accelerate an electron to an energy of 5.6 keV requires 5.6 kilovolts

You wish to observe a virus with a diameter of 20 nm,

which is much too small to observe with an optical

microscope Calculate the voltage required to produce an

electron DeBroglie wavelength suitable for studying this

virus with a resolution of dmin = 2 nm The “f-number” for

an electron microscope is quite large: f/D 100

(Hint: First find required to achieve dmin with the given

f/D Then find E of an electron from .)

object f

electron gun

D

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§ 2 Heisenberg Uncertainty Principle:

2.1 Wave packet and uncertainty:

Wave-like properties of particles (electrons, photons, etc.) reflect afundamental uncertainty in the “knowability” (existence?) of the

particle’s precise location

 For classical waves one can produce a localized “wave packet” by

superposing waves with a range of wave vectors k E.g.:

 k: the spread in wave number

 x: the spread in coordinate (the size

 of the wave packet)

• For wave packets: k.x 1 (see the next slides in more details)

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From the quantum relation between momentum and wavelength p = h/

and the relation k = 2/ p = (h/2).k = ħ.k, where ħh/2 bar”) we have a relation between the spread in the particle’s locations

(“h-x and likely momenta p

ħ(k·x 1)  (ħk)·x ħ  px·x ħThis relation is known as the Heisenberg Uncertainty Principle

To understand the relation k·x 1 we consider the following

example

k → 0

Wave with definite k () monochromatic plane wave

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 Numerical example of

a wavepacket:

• Consider the superposition

of 7 sinusoidal waves with

the frequencies & amplitudes

as below The component

waves have a “spead” in k,

denoted by k

The result has such form

k Nitro PDF Trialwww.nitropdf.com

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• The superposition of an infinite

set of waves with the same “spread”

k has the form which is called

“wave packet”

Wave packet

Note that in the case of an infinite

set of component waves which cover

a continum of k values, the

superposition is a single wave packet

The Fourier mathematical analytics can

provide rigorously the foundation

for this result The wave packet y(x)

is represented by the following integral:

From this integral one can show that k (the “spread” in k) and x (the

“spread” in x) are related through the equation k.x 1

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The meaning of the Heisenberg uncertainty principle: “we cannot knowboth the position and the momentum of a particle simuntaneously withcomplete certainity”.

This principle is of fundamental importance in quantum physics

It means also that in quantum physics there exists not the concept of

a particle’s “path”

Note that this uncertainty is from wave nature of particle, but not fromerrors of experimental measurements

2.2 Uncertainty in macro- and microscopic worlds:

Example 1: A person of mass 60 kg who is moving along the x-axis with

a velocity of 1.5 m/s The uncertainty principle gives

This uncertainty is clearly negligible in the macroscopic world

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Example 2: An electron which is moving with a velocity 2.2x10 6 m/s

(a typical value of electron velocity in atom) We have

This distance is comparable to the size of the atom, so that this

uncertainty must be important for electrons

Conclusion: The Heisenberg uncertainty becomes important only for

microscopic objects In the macroscopic world, this uncertainty is

negligible

2.3 Uncertainty for energy and time:

The periodity of sinusoidal waves is expressed by the function

cos(ωt – kx) By analog to the relation k.x 1 for the pair (k, x), wecan derive the relation ω.t 1 for the pair (ω, t)

Then from E = h h(/  ħ  ħ(ω.t)  ħ, and we have

E.t  ħ The Heisenberg uncertainty for energy and time.The longer lifetime of a state, the smaller is its spread in energy

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§3.The Schrödinger equation:

Having established that matter acts qualitatively like a wave, we want

to be able to make precise quantitative predictions, under given conditions.Usually the “conditions” are specified by giving a potential energy

U(x,y,z) in which the particle is located

E.g., * electron in the coulomb potential of the nucleus

* electron in a molecule

* electron in a solid crystal

* electron in a semiconductor ‘quantum well’

In QM this is notpossible (Why?)

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3.1 Wave function:

 We will see that we can get good predictions (actually, so far they

have never been wrong!!) by assuming that

the state of a particle is described by a “wave function”

(or “probability amplitude”):

 What do we measure? Often it’s: (x,y,z,t)

|(x,y,z,t)| 2 = the probability density (per unit volume) for detecting

a particle near some place (x,y,z), and at some time t

 We need a “wave equation” describing how (x,y,z,t) behaves It should

 be as simple as possible

 make correct predictions

 reduce to the usual classical laws of physics when applied to

“classical” objects (e.g., baseballs)

 (x,y,z,t)

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This equation describes the full time- and space dependence of a

quantum particle in a potential U(x), replacing the classical particledynamics law, a=F/m

 Important feature: Superposition Principle

 The time-dependent SEQ is linear in (a constant times is also asolution), and so the Superposition Principle applies:

If 1 and 2 are solutions to the time-dependent SEQ, then so isany linear combination of 1 and 2

 In 1926, Erwin Schrödinger proposed an equation that described

the time- and space-dependence of the wave function for slow

matter waves (i.e., electrons, protons, NOT photons)

The (1-D) Schrödinger Equation (SEQ)

3.2 The Schrödinger Equation (time-dependent):

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3.3 The (time-independent) Schrödinger Equation:

Before we consider the full time-dependence of states, we will look at aspecial set of states, called “stationary”, which do nothing interesting intime  the probability density ||2 does not change with time

• A state with a definite value of E is stationary:

Since E = h. ħ  is definite  this corresponds to the case ofmonochromatic waves

For monochromatic waves the “t”-dependence is cos(t), or sin(t),

or, more conveniently, exp(-it) = exp(-iEt/ħ)

• In 1 dimension we can write (x,t)= (x).exp(-iEt/ħ) Substituting thisform in 1-D SEQ (see the previous slide), we can separate the “t” variableand obtain

This is one-dimensional Schrödinger Equation for stationary states

(states with a definte energy, or eigenstates)

) ( )

( ) (

) (

2 2

x E

x x

U dx

x d

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 What does the time-independent SEQ represent?

It’s actually not so puzzling…it’s just an expression of a familiar result:

Kinetic Energy (KE) + Potential Energy (PE) = Total Energy (E)

) x ( E )

x ( ) x (

U dx

) x ( d

m

2 2

dx

)x(dm2

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3.4 Complementary conditions for wave functions:

As you see, the Schrödinger Equation is a differential equation of thesecond order in wave function  To solve it and derive physical

consequences, we need use complementary conditions for 

Restricting ourselves in the 1-D time-independent SEQ, = (x),the following conditions must be imposed:

x& d/dx should be

• finite

• single-valued

• continous

What’s a quantum mechanical problem?

For a given potential function U(x), you must

• substitute U(x) in the Schrödinger equation

• identify the boundary conditions for wave function (x)

• find the eigenvalues of energy E (the energy levels)

• find the corresponding eigenstates (the specified wave functions)

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§4 Solutions for some quantum systems:

By some examples you will see how can solve a quantum mechnical

problem and derive physical consequences

4.1 Particle in a potential well:

A potential well is of the following form:

U(x) = 0 when 0 < x < a

∞ when x ≤0 and x ≥a

• In classical mechanics, particle can

move in the 1-D box, and have any energy (a continuum od energy levels)

 How do particle behave in quantum mechanics?

• Inside the well:

The time independent SEQ for the region 0 < x < a (U = 0):



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 where

The general solution of this equation is

• Outside the well: ψ( x ) = 0 everywhere

• Apply the required conditions to ψ(x):

From the single valuedness at x = 0: ψ(x=0) = 0  A = 0

The eigenvalues for k

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The solutions for the problem (the eigenfunctions) are therefore

These are the wave functions which describe the states of a particle inthe 1-D potential well They are “labelled” by interger numbers n.

Now recall the equation , from the eigenvalues for k 

It means that the particle’s energy may have a discrete set of values,called the energy levels of a particle in the potential well

Note that En is proportional to n2.

• Normalization of the eigenfunctions:

Recall that |ψ(x)| 2 = the probability density (per unit volume) of findingthe particle  the total probability must be 1:

( x 2dx

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2 2

2

a

x n B

sin

2)

x

n





These are the normalized

eigenfunctions which describe

the particle’s states in

the well

* Remark: The lowest particle’s

energy must be E1 which

corresponds to n=1.

We can not take n=0,

because in this case ψ  0

over all the space

 It means that there exists

no particle !

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4.2 Tunneling through a potential barrier:

A potential barrier is of the following form:

U(x) = U0 when 0 < x < a

0 when x ≤0 and x ≥a

, 0 )

(

)

1 2

Classically, a particle of total energy E

in the region x < 0 (the region I)

• will remain in (I) as if E < U0

• can move to the region (II) & (III) when E >U0

But the situation is very different in quantum mechanics !

Time independent SEQ:

In the regions A & C:

2 02

E U

m



(We are interested in the case E < U0  k2 is real)

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The solutions have the following form:

, )

Be Ae

, )

De Ce

, )

Ge Fe

The equations for the coefficents:

(the single valuedness of ψat x=0)

(in the region x>a there can not be waves reflected from ∞)

(the single valuedness of ψat x=a)(the single valuedness of dψ/dx at x=0)

G F

k a

ik

Ge Fe

Ce 1  2  2

F k G

k B

ik A

a k a

k a

k

Fe k

Ge k

Ce

2 2

C

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One introduces two quantities which is of physical interest:

R: The coeficient of reflection (the ratio between thesquare of amplitude of the reflected wave and that ofthe incident wave)

T: The coeficient of transmission (the ratio between thesquare of amplitude of the transmitted wave and that ofthe incident wave)

R and T give the probabilities of reflected and transmitted waves,

E U

E

0 0

2

E U

This phenomenon is called barrier penetration or tunneling

Tiêu đề	Quantum Mechanics
Trường học	General Physics III
Chuyên ngành	Optics & Quantum Physics
Thể loại	chapter

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