quantum mechanics v3 slides

8/20/2019 Quantum Mechanics v3 Slides

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Particle in a Box

Finite Square Well

Potential Barrier and Tunneling

Harmonic Oscillator

3D Schrödinger Equation


2/60

Apply the energy eigenfunctions and eigenvalues of an

infinite square well to physical problems.

Show the general solution for the time-dependentSchrödinger equation for an infinite square well potential.

Illustrate how a wavefunction evolves over time in this

system.

Calculate the wavelengths of photons emitted or absorbed

during transitions between energy levels.


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“Infinite Square Well” “Infinite Potential Well”


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= 0 for 0 < <

= ∞elsewhere

()

0

= 2 sin

= ℏ , = 1, 2, 3, …


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Inside the Well ()

0

= − = cos sin

Outside the Well

= 0


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Energy Eigenstates ()

0

= 2 sin Energy Eigenvalues

= ℏ2

()

()()


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Single Separable Solution

, = 2 sin −ℏGeneral Soution

Ψ , =

2 sin −ℏ


8/60

An electron is confined to a box that has a

width of 0.125 nm. The electron makes a

transition from the = 1 to = 4 level byabsorbing a photon. What is the wavelength

of the photon?

3.44 nm


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What is your preferred schedule for the

3rd Long Exam (Saturday May 16, 2015)?

Choose between 7AM-9AM and 3PM-5PM.

Justify your answer.


10/60

What are the energy eigenvalues and

eigenfunctions (time independent) of a

particle in a box of width for a given state? An electron in a box is excited by a photon

with wavelength 415 nm from the ground

state to the first excited state. What is the

width of the box?

= 2 sin =

ℏ2

0.614 nm


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--

-- -

# Ψ(,)Wave Function


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Ψ(,)Wave Function

|Ψ , PDF

Ψ ,

Probability Normalization

−+ Ψ , = 1

Expectation Values

() = −

+ () Ψ , = 1


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Schrödinger Equation

ℏ Ψ(,) = ℏ

2

Ψ , Ψ(, )

Ψ , = = −

ℏ

Separable Solution

ℏ

2

= ()


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ℏ Ψ(,) = ℏ

2

Ψ , Ψ(, )

Ψ , = −

ℏ

General Solution

Probability Amplitude


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Write the appropriate form of the wavefunction of

a finite square well for different regions.

Compare the corresponding energies to the

infinite square well energies.

Calculate the wavelengths of photons emitted or

absorbed during transitions between energy levels.


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()

0

= zero 0 < < elsewhere


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()

0

Inside the Well

= cos sin Outside the Well

= − = 2

ℏ = 2( )

ℏ


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()

0

Region

= Region = cos sin Region = −


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()

0

= 6

= 0.625

= 2.43

= 5.09


20/60

An electron is bound in a square well of width

0.50 nm and depth = 6 . If the electron isinitially in the ground level and absorbs a

photon, what maximum wavelength can the

photon have and still liberate the electron from

the well? 153 nm


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Long Problem Set 3-3 due May 6, 2015 11:59

PM. (12 items LH 35-38)

Take Home Recit due today

3rd Long Exam #truthbetold #realtalk


22/60

A proton is bound in a square well of width 4.0 x 10-15 m.

The depth of the well is six times the ground-level energy

of the corresponding infinite well. If the proton makes a

transition from the ground energy level to the second

excited level by absorbing a photon, find the wavelength of

the photon.

p= 1 673 × 10

−

kg = 0.625 = 2.43 = 5.09 = 6

2.17 x 10-14 m


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Discuss differences in quantum and classical

predictions of some unbound systems.

Calculate for the probability of transmitting a quantum

particle into classically forbidden regions.

Predict how changing the different physical parameters

affect the probability of transmission


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Rectangular Potential Barrier

()

0

=

zero

0 < <

elsewhere


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()

0

Region

= −Region

=

−

Region = − = = 2ℏ = 2( )ℏ


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A particle with energy < originating from ∞ encounters apotential shown by the figure. ()

0

2

∞

Write down the physically correct wavefunctions for regions , , ,and


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()

0

Region

= −Region

=

−

Region = = 2ℏ = 2( )ℏ


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=

= 1 sinh

4

−

when ≪ 1

= 16 1 −

= 2( )ℏ


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A 2.0 eV electron encounters a barrier with

height 5.0 eV and width 1.00 nm. What is the

probability that the electron will tunnel

through the barrier?


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Frustrated Total Internal Reflection

Scanning Tunneling Microscope

http://zotzine.uci.edu/v01/2009_02/images_issue/wilsonho/wilsonhoatom_p090218_03a.jpg

http://researcher.ibm.com/researcher/view_project.php?id=4245


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Problem Set 3-3 due tomorrow May 6,

2015 11:59PM

3rd LE Saturday May 16, 2015 7AM-9AM

Take Home Recit due today


33/60

A particle with energy < originating from ∞ encounters apotential shown by the figure. ()

0

2

∞

Write down the physically correct wavefunctions for regions , , ,and

−

−

sinzero


34/60

An electron with initial kinetic energy 5.0 eV

encounters a barrier with height and width0.60 nm. What is the tunneling probability if

(a) = 7.0 eV; (b) = 9.0 eV; (c) = 13.0 eV5.47 x 10-4 1.81 x 10-5 1.06 x 10-7


35/60

Show the general solution to the time – dependent Schrodinger

equation for a harmonic oscillator

Derive the allowable energies for this system

Compare the classically allowable energies for a quantum

oscillator and a classical oscillator

Calculate the wavelengths of photons emitted or absorbed

during transitions between energy levels


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() = ′

= 12 = 12

Force of a spring

Harmonic Oscillator Potential

http://ffden-2.phys.uaf.edu/211_fall2013.web.dir/Jody_Gaines/Images/Simple%20Harmonic%20Motion.gif


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Harmonic Oscillator Potential

= 12Time Independent


ℏ

2

1

2() = ()


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Energy Eigenfunctions

= ℏ

12!

−/

Hermite Polynomials ()

= ℏ

1

×

−

= 0, 1, 2, 3, …


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Energy Eigenvalues

= 12 ℏ = 0, 1, 2, 3, …

= 1

2ℏ

= 32ℏ = 52ℏ

= 72ℏ


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= ℏ 1

2! −/

= 12 ℏGeneral Solution to the Schrodinger Equation

Ψ , =

−ℏ


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A particle with mass is under the influenceof a harmonic oscillator potential with angular

frequency

.

What is the wavefunction of the particle if it is

found to be in the ground state?

What is the wavefunction of the particle if it is

found to be in the first excited state?

Ψ , = ℏ

/

−

ℏ −

Ψ , = ℏ

12

ℏ

−ℏ

−


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A particle with mass in a quantumharmonic oscillator vibrates with an angular

frequency of . The particle is prepared in astate that has a 50-50 chance to be found in

either the ground or first excited state.

What is the wave function of this state

assuming all probability amplitudes are real?

Ψ , = 12 ℏ−

ℏ − 12 ℏ

12 ℏ −

ℏ −


43/60

The ground state energy of a harmonic

oscillator is 5.60 eV. If the oscillator

undergoes a transition from its = 3 to = 2 level by emitting a photon, what isthe wavelength of the photon?

= 111 nm


44/60

An atom in a crystal vibrates in simple

harmonic motion with angular frequency equal

to 1.80 x 1013

rad/s.

Find

(a) the ground state energy of the atom

(b) the wavelength of emitted photon when the

transition = 4 to = 2 occurs.9.49 x 10

-22

J or 5.92 x 10-3

eV

5.24 x 10-5 m


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Show how degeneracy arises in quantum systems of more than one

dimension.

Generate the possible quantum states of a system by listing down

the corresponding quantum numbers.

Solve for the energy levels and energy eigenfunctions of a particle

in a 3-D box and determine the degeneracy of each level.

Solve for the energy levels and energy eigenfunctions of a particle

in 3-D in a harmonic potential and determine the degree of

degeneracy of each level.

Calculate the wavelengths of photons emitted or absorbed during

transitions between energy levels.


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ℏ2 Ψ , , , , , , Ψ , , , = ℏ Ψ , , ,

Separable SolutionΨ ,, , = ,, ()

ℏ

2 , , (,, )(,, ) = (,, )

Ψ , , , = , , −ℏTime Independent 3D Schrödinger Equation


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x

y

z

= 0 inside = ∞ outside Inside the Box ℏ

2

=

Separable Solutions

, , = () ℏ2

= () ℏ

2 = ()

ℏ

2

= ()

=


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y

z

= 0 inside = ∞ outside

= 2 sin

,, , , = ()

= 2 sin

= 2 sin

=

ℏ

2 =

ℏ2

= ℏ2

= ℏ

2

Energy Eigenstates

Energy Eigenvalues


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Consider a particle of mass in a 3D box withdimensions = = = What is the ground energy level of the particle?

What is the wavefunction of a particle in the ground

state?

What is the first excited energy level of the particle?

How many states correspond in the first excited

energy level?

3

Ψ , = 2 sin

2 sin

2 sin

−/ℏ

6, 3 states (1,1,2) (1,2,1) (2,1,1)


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Consider a particle of mass in a 3D boxwith dimensions

slightly < slightly <

List down the quantum numbers of the state

corresponding to the lowest 5 energy levels.


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Ground State – (1,1,1)

1st Excited – (1,1,2)

2nd Excited – (1,2,1)

3rd Excited – (2,1,1)

4th Excited – (1,2,2)


52/60

A particle is in a three-dimensional box with

= = 2.List the quantum numbers (, , )

corresponding to the lowest 5 energy levels.

Write the energy of each level in terms of theground state of a 1D infinite square well withwidth and indicate the degeneracy of each level.


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Quantum #’s E Deg

(1,1,1)

321 or none

(2,1,1) (1,2,1) 2.25 2(2,2,1) 3 1 or none

(3,1,1) (1,3,1) 3.5 2(3,2,1) (2,3,1)

4.252


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, , = 12 12 12

, , = 12

12

12

= 1

2 ℏ 1

2 ℏ 1

2 ℏ

Energy Eigenvalues

Energy Eigenstates

,, , , = ()


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An isotropic harmonic oscillator has the potential energy function

= 12 = 12( )

A particle is subjected to this 3D isotropic harmonic oscillator.

(a) What quantum numbers , , correspond to the groundstate of the particle? What is the energy of the ground state?

(b) List all down all the possible quantum numbers , , thatcorresponds to the first & second excited energy level. Find the first

& second excited energy, as well as the degree of degeneracy.


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Quantum #’s E Deg

(0,0,0) 3

2ℏ1 or none

(0,0,1) (0,1,0)(1,0,0)

52ℏ 3(1,1,0) (1,0,1)(0,1,1) (0,0,2)(0,2,0) (2,0,0)

7

2ℏ 6


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An isotropic harmonic oscillator has the potential energy function

= 12 = 12( )

A particle is subjected to this 3D isotropic harmonic oscillator.

List down all quantum numbers

, , correspond to the third

and fourth excited state. Compute for the corresponding energy and

degeneracy of each level.


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Quantum #’s E Deg

(0,0,3) (0,3,0) (3,0,0) (1,0,2)(1,2,0) (0,1,2) (2,1,0) (0,2,1)

(2,0,1) (1,1,1)

92ℏ 10(0,0,4) (0,4,0) (4,0,0) (1,0,3)(1,3,0) (0,1,3) (3,1,0) (0,3,1)

(3,0,1) (2,0,2) (2,2,0) (0,2,2)(1,1,2) (1,2,1) (2,1,1)

52ℏ

15


59/60

An electron is enclosed in a 3D cube with

sides equal to 0.10 nm. What is the

wavelength of a photon the electron must

absorbed in order to raise it’s energy from

the ground energy level to the first excitedlevel?

= 1.10 × 10−8 m


60/60

An atom in a 3D crystal vibrates in an isotropic

harmonic potential with an associated angular

frequency of 1.80 x 1013 rad/s for all dimensions.

Find the wavelength of a photon that the atom

must absorb in order for the atom to transition

from the ground state to the first excited state.−

quantum mechanics v3 slides

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