Chapter 5

Quantum Mechanics

5 Quantum Mechanics

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5 Quantum Mechanics

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• The Bohr theory of the atom has a number of severe limitations.

• It applies only to hydrogen and one-electron ions such as He

and Li

• It cannot explain why certain spectral lines are more intense than others (that is,

why certain transitions between energy levels have greater probabilities of

occurrence than others).

• It cannot account for the observation that many spectral lines actually consist of

several separate lines whose wavelengths differ slightly.

• Perhaps most important, it does not permit us to obtain an understanding of how

individual atoms interact with one another to endow macroscopic aggregates of

matter with the physical and chemical properties we observe.

• A more general approach to atomic phenomena is required.

• Such an approach was developed in 1925 and 1926 by Erwin

Schrödinger, Werner Heisenberg, Max Born, Paul Dirac, and others

under the name of quantum mechanics.

5. 1 Quantum Mechanics

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• Classical mechanics is an approximation of quantum mechanics

• The fundamental difference between classical (or Newtonian)

mechanics and quantum mechanics lies in what they describe.

• In classical mechanics, the future history of a particle is completely

determined by its initial position and momentum together with the

forces that act upon it.

• Quantum mechanics also arrives at relationships between observable

quantities, but the uncertainty principle suggests that the nature of an

observable quantity is different in the atomic level.

5. 1 Quantum Mechanics: Wave Function

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• The quantity with which quantum mechanics is concerned is the wave

function Ψ of a body.

• The linear momentum, angular momentum, and energy of the body are

quantities that can be established from Ψ.

• The problem of quantum mechanics is to determine Ψ for a body when

its freedom of motion is limited by the action of external forces.

• Wave functions are usually complex with both real and imaginary

parts

(Wave function)

where A and B are real functions.

• A probability, however, must be a positive real quantity. The

probability density |Ψ|

for a complex is therefore taken as the product

of Ψ and its complex conjugate Ψ* which is Ψ*Ψ.

• The complex conjugate of any function is obtained by replacing i by -i

wherever it appears in the function.

• Since i

=-1; |Ψ|

=Ψ*Ψ is always a positive real quantity.

(Complex conjugate)

5. 1 Quantum Mechanics: Normalization

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• Since |Ψ|

is proportional to the probability density P of finding the

body described by Ψ, the integral of |Ψ|

over all space must be finite -

the body is somewhere.

• If the particle does not exist.

• It is usually convenient to have |Ψ|

be equal to the probability density

P of finding the particle described by Ψ, rather than merely be

proportional to P.

• If |Ψ|

is to equal P, then it must be true that

• A wave function that obeys Eq. (5.1) is said to be normalized.

• Every acceptable wave function can be normalized by multiplying it by

an appropriate constant.

(5.1 Normaliation)

5.1 Quantum Mechanics: Well-Behaved Wave Functions

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• Only wave functions with the properties below can yield physically

meaningful results when used in calculations, so only such “well-

behaved” wave functions are admissible as mathematical

representations of real bodies.

• To summarize:

1. Ψ must be continuous and single-valued everywhere.

2. Ψ/x, Ψ/y, Ψ/z must be continuous and single-valued

everywhere.

3. Ψ must be normalizable, which means that Ψ must go to 0 as

x→ , y → , z → in order that  Ψ|

dV over all space be a

finite constant.

(Probabilty)

• For a particle restricted to motion in the x direction, the probability

of finding it between x

and x

is given by

5.2 The Wave Equation

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• It can have a variety of solutions, including complex ones.

• Schrödinger’s equation, which is the fundamental equation of quantum

mechanics in the same sense that the second law of motion is the

fundamental equation of Newtonian mechanics, is a wave equation in

the variable Ψ.

(5.3 Wave equation)

• Solutions of the wave equation may be of many kinds, reflecting the

variety of waves that can occur.

Figure 5.1 Waves in the xy plane traveling in

the x direction along a stretched string lying

on the x axis.

All solutions must be of the form

where F is any function that can be

differentiated.

• The solutions F(t-x/v) represent waves

traveling in the +x-direction,

• and the solutions F(t+x/v) represent waves

traveling in the -x direction.

5.2 The Wave Equation

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• Let us consider the wave equivalent of a “free particle”, which is a

particle that is not under the influence of any forces and therefore

follow a straight path at constant speed.

• This wave is described by the general solution of Eq. (5.3) for

undamped (that is, constant amplitude A), monochromatic (constant

angular frequency) harmonic waves in the x direction, namely

In this formula y is a complex quantity, with both real and imaginary

parts.

5.3 Schrödinger’s Equation: Time-Dependent Form

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• A basic physical principle that cannot be derived from anything else.

• In quantum mechanics, the wave function Ψ corresponds to the wave

variable y of wave motion in general.

• However, Ψ, unlike y, is not itself a measurable quantity and may

therefore be complex. For this reason, we assume that for a particle

moving freely in the +x-direction is specified by

Replacing  in the above formula by 2 and v by  gives

• This is convenient since we already know what  and  are in terms of

the total energy E and momentum p of the particle being described by

Ψ. Because we have

(5.9 Free

particle)

• Equation (5.9) describes the wave equivalent of an unrestricted particle

of total energy E and momentum p moving in the +x- direction.

• The expression for the wave function Ψ given by Eq. (5.9) is correct

only for freely moving particles.

5.3 Schrödinger’s Equation: Time-Dependent Form

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• However, we are most interested in situations where the motion of a

particle is subject to various restrictions.

• An important concern, for example, is an electron bound to an atom by

the electric field of its nucleus.

• What we must now do is obtain the fundamental differential equation

for Ψ, which we can then solve for in a specific situation.

• This equation is Schrödinger’s equation.

• We begin by differentiating Eq. (5.9) for Ψ twice with respect to x,

which gives

differentiating Eq. (5.9) once with respect to t gives

(5.10)

(5.11)

5.3 Schrödinger’s Equation: Time-Dependent Form

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• At speeds small compared with that of light, the total energy E of a

particle is the sum of its kinetic energy p

2m and its potential energy

U, where U is in general a function of position x and time

(5.12)

(5.14 Time dependent

Schrödinger’s equation in 1D)

• The function U represents the influence of the rest of the universe on

the particle.

• Of course, only a small part of the universe interacts with the particle

to any extent;

• for instance, in the case of the electron in a hydrogen atom, only the

electric field of the nucleus must be taken into account.

• Multiplying both sides of Eq. (5.12) by the wave function Ψ.

• Now we substitute for E Ψ and p

Ψ from Eqs. (5.10) and (5.11) to

obtain the time dependent form of Schrödinger’s equation:

5.3 Schrödinger’s Equation: Time-Dependent Form

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where the particle’s potential energy U is some function of x, y,

z, and t.

• Any restrictions that may be present on the particle’s motion

will affect the potential energy function U.

• In three dimensions the time-dependent form of

Schrödinger’s equation is

Ernwin

Schrödinger

(1887–1961)

Nobel Prize in

Physics in 1933

• Once U is known, Schrödinger’s equation may be solved for the wave

function Ψ of the particle, from which its probability density |Ψ|

may

be determined for a specified x, y, z, t.

Schrödinger’s equation cannot be derived from other basic principles of

physics; it is a basic principle in itself.

(5.15 Time dependent

Schrödinger’s equation in 3D)

5.4 Linearity and Superposition

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• Wave functions add, not probabilities.

• An important property of Schrödinger’s equation is that it is linear in

the wave function: the equation has terms that contain and its

derivatives but no terms independent of or that involve higher powers

of or its derivatives.

• As a result, a linear combination of solutions of Schrödinger’s equation

for a given system is also itself a solution.

• If Ψ

and Ψ

are two solutions (that is, two wave functions that satisfy the

equation), then Ψ= a

is also a solution, where a

and a

are constants.

Superposition principle.

• We conclude that interference effects can occur for wave functions just

as they can for light, sound, water, and electromagnetic waves.

• Let us apply the superposition principle to the diffraction of an electron

beam.

5.4 Linearity and Superposition

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• Figure 5.2a shows a pair of slits through which a parallel beam of

monoenergetic electrons pass on their way to a viewing screen.

Figure 5.2 (a) Arrangement of double-slit experiment. (b) The

electron intensity at the screen with only slit 1 open. (c) The electron

intensity at the screen with only slit 2 open. (d) The sum of the

intensities of (b) and (c). (e) The actual intensity at the screen with

slits 1 and 2 both open. The wave functions Ψ

and Ψ

add to

produce the intensity at the screen, not the probability densities |Ψ1|

and |Ψ2|

• If slit 1 only is open, the result is the intensity variation shown in Fig. 5.2b that

corresponds to the probability density P=|Ψ

= Ψ

• If slit 2 only is open, as in Fig. 5.2c, the corresponding probability density is

P=|Ψ

= Ψ

• We might suppose that opening both slits would give an electron intensity variation

described by P

+ P

, as in Fig. 5.2d

• However, this is not the case because in quantum mechanics wave functions add,

not probabilities.

• Instead the result with both slits open is as shown in Fig. 5.2e, the same pattern of

alternating maxima and minima that occurs when a beam of monochromatic light

passes through the double slit of Fig. 2.4. Superposition of the wave functions.

5.5 Expectation Values

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• How to extract information from a wave function.

• Once Schrödinger’s equation has been solved for a particle in a given

physical situation, the resulting wave function Ψ(x, y, z, t) contains all

the information about the particle (that is permitted by the uncertainty

principle).

• Let us calculate the expectation value <x> of the position of a particle

confined to the x axis that is described by the wave function Ψ(x, t).

• This is the value of x we would obtain if we measured the positions

of a great many particles described by the same wave function at

some instant t and then averaged the results.

• What is the average position x of a number of identical particles

distributed along the x axis in such a way that there are N

particles at

, N

particles at x

, and so on? The average position in this case is the

same as the center of mass of the distribution, and so

5.5 Expectation Values

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• When we are dealing with a single particle, we must replace the

number N

of particles at x

by the probability P

that the particle be

found in an interval dx at x

• This probability is P

=| Ψ

dx where Ψ

is the particle wave function

evaluated at x=x

. Making this substitution and changing the

summations to integrals, we see that the expectation value of the

position of the single particle is

• If Ψ is a normalized wave function, the denominator of Eq. (5.18)

equals the probability that the particle exists somewhere between

x=- and x= therefore has the value 1. In this case

(5.19 Expectation value for position)

 The same procedure as that followed above can be used to obtain the

expectation value G(x) of any quantity-for instance, potential energy

U(x)-that is a function of the position x of a particle described by a

wave function Ψ. The result is

(5.20 Expectation value for G)

5.5 Expectation Values

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Example 5.2

A particle limited to the x axis has the wave function Ψ=ax between x=0

and x=1; Ψ= 0 elsewhere. (a) Find the probability that the particle can be

found between x=0.45 and x=0.55. (b) Find the expectation value <x> of

the particle’s position.

5.6 Operators

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• Another way to find expectation values.

• A hint as to the proper way to evaluate <p> and <E> comes from

differentiating the free particle wave function Ψ=A e

(-i/h(Et-px))

with

respect to x and to t. We find that

(5.21)

(5.22)

 An operator tells us what operation to carry out on the quantity that

follows it.

 Thus, the operator E instructs us to take the partial derivative of what

comes after it with respect to t and multiply the result by .

 It is customary to denote operators by using a caret, so that ˆp is the

operator that corresponds to momentum p and ˆE is the operator that

corresponds to total energy E.

 From Eqs. (5.21) and (5.22) these operators are

(5.23 Momentum operator)

(5.24 Total-energy operator)

They are entirely

general results whose

validity is the same as

that of Schrödinger’s

equation.

5.6 Operators

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• Replace the equation E=KE+U for the total energy of a particle with

the operator equation we have

(5.25)

which is Schrödinger’s equation. Postulating Eqs. (5.23) and (5.24) is

equivalent to postulating Schrödinger’s equation.

5.6 Operators: Operators and Expectation Values

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(5.30 Expectation value of an operator)

• Because p and E can be replaced by their corresponding operators in

an equation, we can use these operators to obtain expectation values

for p and E. Thus the expectation values for p and E are

 Every observable quantity G characteristic of a physical system may

be represented by a suitable quantum-mechanical operator ˆG. To

obtain this operator, we express G in terms of x and p. If the wave

function Ψ of the system is known, the expectation value of G(x, p) is

5.7 Schrödinger’s Equation: Steady-State Form

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(5.31)

• Eigenvalues and eigenfunctions.

• In a great many situations, the potential energy of a particle does not

depend on time explicitly;

• the forces that act on it, and hence U, vary with the position of the

particle only.

• Then Schrödinger’s equation may be simplified by removing all

reference to t.

Ψ is now the product of a time-dependent function exp(-(iE/h)t) and a

position dependent function ψ.

Substituting the Ψ of Eq. (5.31) into the time-dependent form of

Schrödinger’s equation and dividing through the common exponential

factor, we find that

(5.32 Steady-state Schrödinger

equation in one dimension)

Equation (5.32) is the steady-state form of Schrödinger’s equation.

5.7 Schrödinger’s Equation: Steady-State Form & Eigenvalues and Eigenfunctions

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• In three dimensions, the steady-state form of Schrödinger’s

equation is

 An important property of Schrödinger’s steady-state equation is that, if

it has one or more solutions for a given system, each of these wave

functions corresponds to a specific value of the energy E.

 Thus, energy quantization appears in wave mechanics as a natural

element of the theory.

 The values of energy E

for which Schrödinger’s steady-state equation

can be solved are called eigenvalues and the corresponding wave

functions ψ

are called eigenfunctions.

 The discrete energy levels of the hydrogen atom are an example of a

set of eigenvalues.

5.7 Schrödinger’s Equation: Operators and Eigenvalues

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 An important example of a dynamical variable other than total energy

that is found to be quantized in stable systems is angular momentum L.

In the case of the hydrogen atom, eigenvalues of the magnitude of the

total angular momentum:

• In the hydrogen atom, the electron’s position is not quantized. So

that we must think of the electron as being present in the vicinity of the

nucleus with a certain probability |ψ|

per unit volume but with no

predictable position or even orbit in the classical sense.

• The condition that a certain dynamical variable G be restricted to the

discrete values G

(G be quantized) is that the wave functions ψ

the system be such that

(5.34 Eigenvalue equation)

where ˆG is the operator that corresponds to G and each G

is a real

number.

 If measurements of G are made on a number of identical systems

(eigenfunction ψ

), each measurement will yield the single value

5.5 Expectation Values

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Example 5.3

An eigenfunction of the operator d

/dx

is ψ=e

(2x)

. Find the

corresponding eigenvalue.

5.7 Schrödinger’s Equation: Operators and Eigenvalues

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 The total-energy operator ˆE can also be written as

(5.35 Hamiltonian operator)

and is called the Hamiltonian operator. The steady-state

Schrödinger equation can be written simply as

(5.36 Schrödinger's equation)

Table 5.1 Lists the operators that correspond to various

observable quantities.

5.8 Particle in a Box

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• We may specify the particle’s motion by saying that

it is restricted to traveling along the x axis between

x=0 and x=L by infinitely hard walls.

• A particle does not lose energy when it collides with

such walls, so that its total energy stays constant.

Figure 5.4 A square potential well

with infinitely high barriers at each

end corresponds to a box with

infinitely hard walls.

• Potential energy U of the particle is infinite on both

sides of the box, while U is a constant -say 0 for convenience- on the

inside (Fig. 5.4).

• Because the particle cannot have an infinite amount of energy, it

cannot exist outside the box, and so its wave function ψ is 0 for x≤0

and x≥L.

• How boundary conditions and normalization determine wave

functions.

• The simplest quantum-mechanical problem is that of a particle trapped

in a box with infinitely hard walls.

5.8 Particle in a Box: Eigenvalues

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• Our task is to find what ψ is within the box (btw x=0 and x=L). Within

the box Schrödinger’s equation becomes

(5.37)

since U=0 there. Equation (5.37)

has the solution

(5.38)

A and B are constants to be

evaluated

• This solution is subject to the boundary conditions:

• ψ=0 for x=0 and for x=L. Since cos 0=1, the second term cannot

describe the particle because it does not vanish at x=0. Hence, B=0.

• Since sin0=0, the sine term always yields ψ=0 at x=0, as required,

but ψ will be 0 at x=L only when

(5.39)

• From Eq. (5.39), Energy of the particle can have only certain values

(eigenvalues). These eigenvalues, constituting the energy levels of the

system, are found by solving Eq. (5.39) for E

, which gives

(5.40 Particle in a box)

5.8 Particle in a Box: Eigenfunctions

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• The wave functions (eigenfunctions) of a particle in a box whose

energies are E

are, from Eq. (5.38) with B=0

(5.41)

substituting Eq. (5.40) for E

gives

(5.42)

for the eigenfunctions corresponding to the

energy eigenvalues En.

• With the help of the trigonometric identity sin

θ=1/2(1-cos2θ) we find

that

(5.43)

5.8 Particle in a Box: Normalization

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• To normalize ψ we must assign a value to A such that |ψ

dx is equal

to the probability Pdx of finding the particle between x and x+dx,

rather than merely proportional to Pdx. If |ψ

dx is to equal P dx, then

it must be true that

(5.44)

Comparing Eqs. (5.43) and (5.44), we see that the wave

functions of a particle in a box are normalized if

(5.45)

The normalized wave functions of the particle are

therefore

(5.46 Particle in a box)

5.8 Particle in a Box

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• The normalized wave functions ψ

, ψ

, and ψ

together

with the probability densities |ψ

, |ψ

, and |ψ

are

plotted in Fig. 5.5.

• Although ψ

may be negative as well as positive, |ψ

is never negative.

• Since ψ

is normalized, its value (|ψ

) at a given x is

equal to the probability density of finding the particle

there.

• In every case |ψ

=0 at x=0 and x=L, the boundaries of

the box.

Figure 5.5 Wave

functions and

probability densities of

a particle confined to a

box with rigid walls.

5.8 Particle in a Box

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Example 5.4

Find the probability that a particle trapped in a box L wide can be found

between 0.45L and 0.55L for the ground and first excited states.

Figure 5.6 The probability P

x1,x2

of finding a particle in the box of

Fig. 5.5 between x

=0.45L and

=0.55L is equal to the area

under the |ψ|

curves between

these limits.

5.8 Particle in a Box

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Example 5.5

Find the expectation value <x> of the position of a particle trapped in a

box L wide.

5.9 Finite Potential Well

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• The wave function penetrates the walls, which lowers the energy levels.

• Potential energies are never infinite in the real world, and the box with

infinitely hard walls (particle in a box) has no physical counterpart.

• However, potential wells with barriers of finite height certainly do

exist.

• Figure 5.7 shows a potential well with square

corners that is U high and L wide and contains a

particle whose energy E is less than U

• According to classical mechanics, when the

particle strikes the sides of the well, it bounces

off without entering regions I and III.

Figure 5.7 A square potential well

with finite barriers. The energy E of

the trapped particle is less than the

height U of the barriers.

• In quantum mechanics, the particle also bounces back and forth, but

now it has a certain probability of penetrating into regions I and III

even though E<U.

5.9 Finite Potential Well

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• In regions I and III Schrödinger’s steady-state equation is

which we can rewrite in the more convenient form

(5.53)

The solutions to

Eq. (5.53) are real

exponentials:

• Both ψ

and ψ

III

must be finite everywhere. (e



→0)

Since e

-ax

→ as x →- and e

→ as x → , the

coefficients D and F must therefore be 0. Hence we have

These wave functions decrease exponentially inside the barriers at the

sides of the well.

• In regions II (within the well). Schrödinger’s equation is the same as

Eq. (5.37) and its solution is

(5.59)

Here, ψ

=C at x=0 and ψ

=G at x =L, so both the sine and cosine

solutions of Eq. (5.59) are possible.

5.9 Finite Potential Well

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• For either solution, both ψ and dψ/dx must be

continuous at x=0 and x=L:

• The wave functions inside and outside each side of

the well must not only have the same value where

they join.

• But also the same slopes, so they match up perfectly.

• When these boundary conditions are taken into

account, the result is that exact matching only occurs

for certain specific values E

of the particle energy.

• The complete wave functions and their probability

densities are shown in Fig. 5.8.

Figure 5.8 Wave functions

and probability densities

of a particle in a finite

potential well. The

particle has a certain

probability of being found

outside the wall.

5.11 Harmonic Oscillator

• Its energy levels are evenly spaced.

• Harmonic motion takes place when a system of some kind vibrates

about an equilibrium configuration. The system may be

• an object supported by a spring,

• floating in a liquid,

• a diatomic molecule,

• an atom in a crystal lattice … on all scales of size.

• The condition for harmonic motion is the presence of a restoring force

that acts to return the system to its equilibrium configuration when it is

disturbed.

(Hooke’s law)

This relationship is customarily called Hooke’s law. From the second

law of motion, F=ma, we have

(5.62 Harmonicc oscillator)

5.11 Harmonic Oscillator

(5.64 Frequency of harmonic oscillator)

• There are various ways to write the solution to Eq. (5.62). A common

one is

ν is the frequency of the oscillations and A is their amplitude. The value

of , the phase angle, depends upon what x is at the time t=0 and on the

direction of motion then.

• The importance of the simple harmonic oscillator in both classical

and modern physics lies not in the strict adherence of actual restoring

forces to Hooke’s law, which is seldom true, but in the fact that these

restoring forces reduce to Hooke’s law for small displacements x.

• As a result, any system in which something executes small vibrations

about an equilibrium position behaves very much like a simple

harmonic oscillator.

5.11 Harmonic Oscillator

• The potential-energy function U(x) that corresponds to a Hooke’s law

force may be found by calculating the work needed to bring a particle

from x=0 to x=x against such a force. (see Figure 5.10)

Figure 5.10 The potential energy of a

harmonic oscillator is proportional to x

where x is the displacement from the

equilibrium position. The amplitude A of the

motion is determined by the total energy E of

the oscillator, which classically can have any

value.

• Three quantum mechanical modifications to

this classical picture:

1.The allowed energies will not form a

continuous spectrum but instead a discrete

spectrum of certain specific values only.

2.The lowest allowed energy will not be E=0

but will be some definite minimum E=E

3.There will be a certain probability that the

particle can penetrate the potential well it is

in and go beyond the limits of -A and +A.

5.11 Harmonic Oscillator: Energy Levels

• Schrödinger’s equation for the harmonic oscillator is, with U=1/2kx

Figure 5.11 Potential wells and energy levels

of (a) a hydrogen atom, (b) a particle in a box,

and (c) a harmonic oscillator. In each case the

energy levels depend in a different way on the

quantum number n. Only for he harmonic

oscillator are the levels equally spaced.

(5.70 Energy levels of harmonic oscillator)

The energy of a harmonic oscillator is thus quantized in

steps of hν. Note that when n=0,

(5.71 Zero point energy)

• This value is called the zero-point energy because a

harmonic oscillator in equilibrium with its surroundings

would approach an energy of E=E

and not E=0 as the

temperature approaches 0 K.

• Figure 5.11 is a comparison of the energy levels of a

harmonic oscillator with those of a hydrogen atom and of

a particle in a box with infinitely hard walls.

• The shapes of the respective potential-

energy curves are also shown.

• The spacing of the energy levels is constant

only for the harmonic oscillator.

5 Solved Problems

1. Show that Ae

i(kx-wt)

satisfies the time-dependent Schrödinger wave

equation.

5 Solved Problems

2. The initial wavefunction of a particle is given as

ψ(x,0)=C e

(-|x|/x0),

where C and x

are constants.

Sketch of the function is given.

a) Find C in terms of x

such that ψ(x,0) is

normalized.

b) Calculate the probability that the particle will

be found in the interval -x

≤ x ≤

5 Solved Problems

3. A small object of mass 1.00 mg is confined to move between two

rigid walls separated by 1.00 cm.

a) Calculate the minimum speed of the object.

b) If the speed of the object is 3.00 cm/s, find the corresponding

value of n.

5 Solved Problems

4. An electron is trapped in a one-dimensional region of length

1.00x10

-10

m (a typical atomic diameter).

a) Find the energies of the ground state and first two excited states.

b) How much energy must be supplied to excite the electron from

the ground state to the second excited state?

c) From the second excited state, the electron drops down to the

first excited state. How much energy is released in this process?

d) In the first excited state, what is the probability of finding the

electron between x=0 and x=0.025 nm?

5 Solved Problems

5. An atom can be viewed as a number of electrons

moving around a positively charged nucleus, where

the electrons are subject mainly to the Coulombic

attraction of the nucleus (which actually is partially

“screened” by the intervening electrons). The

potential well that each electron “sees” is sketched

in Figure. Use the model of a particle in a box to estimate the energy (in

eV) required to raise an atomic electron from the state n=1 to the state

n=2, assuming the atom has a radius of 0.100 nm.

5 Solved Problems

6. A particle is known to be in the ground state of an infinite square

well with length L. Calculate the probability that this particle will be

found in the middle half of the well, that is, between x=L/4 and

x=3L/4.

5 Solved Problems

7. An electron is bound to a region of space by a springlike force with

an effective spring constant of k = 95.7 eV/nm

a) What is its ground-state energy?

b) How much energy must be absorbed for the electron to jump

from the ground state to the second excited state?

5 Quantum Mechanics

Additional Materials

5.8 Particle in a Box

5.10 Tunnel Effect

• A particle without the energy to pass over a

potential barrier may still tunnel through it

• Although the walls of the potential well of

Fig. 5.7 were of finite height, they were

assumed to be infinitely thick.

– As a result, the particle was trapped

forever even though it could penetrate

the walls.

• We next look at the situation of a particle

that strikes a potential barrier of height U,

again with E<U, but here the barrier has a

finite width (Fig. 5.9).

Figure 5.9 When a particle of energy E< U

approaches a potential barrier, according to

classical mechanics the particle must be reflected.

In quantum mechanics, the de Broglie waves that

correspond to the particle are partly reflected and

partly transmitted, which means that the particle

has a finite chance of penetrating the barrier.

• What we will find is that the particle has a certain probability -not

necessarily great, but not zero either- of passing through the barrier

and emerging on the other side.

5.10 Tunnel Effect

• The particle lacks the energy to go over the top of the barrier, but it can

nevertheless tunnel through it, so to speak.

• Not surprisingly, the higher the barrier and the wider it is, the less the

chance that the particle can get through.

• The tunnel effect actually occurs, notably in the case of the alpha

particles emitted by certain radioactive nuclei.

• An alpha particle whose kinetic energy is only a few MeV is able

to escape from a nucleus whose potential wall is perhaps 25 MeV

high.

• The probability of escape is so small that the alpha particle might

have to strike the wall 1038 or more times before it emerges, but

sooner or later it does get out.

• Tunneling also occurs in the operation of certain semiconductor diodes

in which electrons pass through potential barriers even though their

kinetic energies are smaller than the barrier heights.

5.10 Tunnel Effect: Scanning Tunneling Microscope

• The ability of electrons to tunnel through a potential

barner is used in an ingenious way in the scanning

tunneling microscope (STM) to study surfaces on an

atomic scale of size.

• In an STM, a metal probe with a point so fine that its tip is a

single atom is brought close to the surface of a conducting or

semiconducting material.

• Normally even the most loosely bound electrons in an atom on a

surface need several electron-volts of energy to escape -this is the

work function.

• However, when a voltage of only 10 mV or so is applied between

the probe and the surface, electrons can tunnel across the gap

between them if the gap is small enough, a nanometer or two.

• What is done is to move the probe across the surface in a series of

closely spaced back-and-forth scans.

5.10 Tunnel Effect: Scanning Tunneling Microscope

• The height of the probe is continually adjusted to give a constant

tunneling current, and the adjustments are recorded so that a map

of surface height versus position is built up.

• Such a map is able to resolve individual atoms on a surface.

• Actually, the result of an STM scan is not a true topographical

map of surface height but a contour map of constant electron

density on the surface. This means that atoms of different

elements appear differently.

• Although many biological materials conduct electricity, they do so by the flow of

ions rather than of electrons and so cannot be studied with STMs.

• The atomic force microscope (AFM) can be used on any surface, although with

somewhat less resolution than an STM.

• In an AFM, the sharp tip of a fractured diamond presses gently against the

atoms on a surface.

• A spring keeps the pressure of the tip constant, and a record is made of the

deflections of the tip as it moves across the surface.

• The result is a map showing contours of constant repulsive force between the

electrons of the probe and the electrons of the surface atoms.