CHAPTER 6 300 fall 19...Β Β· 2019-10-21Β Β· Fall 2018 Prof. Sergio B. Mendes 9 πβ , =β β2 2...
Transcript of CHAPTER 6 300 fall 19...Β Β· 2019-10-21Β Β· Fall 2018 Prof. Sergio B. Mendes 9 πβ , =β β2 2...
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Quantum Mechanics II
Fall 2018 Prof. Sergio B. Mendes 1
CHAPTER 6
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Topics
Fall 2018 Prof. Sergio B. Mendes 2
6.1 The SchrΓΆdinger Wave Equation
6.2 Expectation Values
6.4 Finite Square-Well Potential
6.3 Infinite Square-Well Potential
6.5 Three-Dimensional Infinite-Potential Well
6.6 Simple Harmonic Oscillator
6.7 Barriers and Tunneling
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6.1 SchrΓΆdinger Equation
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Erwin SchrΓΆdinger (1887β1961) was an Austrian who worked at several European universities before fleeing Nazism in 1938 and accepting a position at the University of Dublin, where he remained until his retirement in 1956.
His primary work on the wave equation was performed during the period he was in Zurich from 1920 to 1927.
SchrΓΆdinger worked in many fields including philosophy, biology, history, literature, and language.
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Free Particle: π = 0
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πΈ = πΎ + π
π π₯, π‘ = π΄ ππ π π₯ β π π‘ + π π π₯, π‘ 2 = π΄ 2
πΈ = β π = β π π =β
π= β π
=π π£2
2+ 0 =
π2
2 π
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The Total Energy:
Fall 2018 Prof. Sergio B. Mendes 5
ππ π₯, π‘
ππ‘
π π₯, π‘ = π΄ ππ π π₯ β π π‘ + π
= β π π π π₯, π‘= β π π π΄ ππ π π₯ β π π‘ + π
πΈ = β π =β
β π
1
π π₯, π‘
ππ π₯, π‘
ππ‘
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The Kinetic Energy:
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π π₯, π‘ = π΄ ππ π π₯ β π π‘ + π
ππ π₯, π‘
ππ₯= π π π΄ ππ π π₯ β π π‘ + π = π π π π₯, π‘
π2π π₯, π‘
ππ₯2=π
ππ₯
ππ π₯, π‘
ππ₯=π
ππ₯π π π π₯, π‘ = β π2π π₯, π‘
πΎ =π2
2 π=
β2
2 ππ2 = β
β2
2 π
1
π π₯, π‘
π2π π₯, π‘
ππ₯2
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Bringing the pieces together for a free particle:
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β π =β2
2 ππ2
β
β π
1
π π₯, π‘
ππ π₯, π‘
ππ‘= β
β2
2 π
1
π π₯, π‘
π2π π₯, π‘
ππ₯2
πΈ = πΎ
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In the presence of a potential π:
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β π =β2
2 ππ2 + π
β
β π
1
π π₯, π‘
ππ π₯, π‘
ππ‘= β
β2
2 π
1
π π₯, π‘
π2π π₯, π‘
ππ₯2+ π π₯, π‘
π βππ π₯, π‘
ππ‘= β
β2
2 π
π2π π₯, π‘
ππ₯2+ π π₯, π‘ π π₯, π‘
πΈ = πΎ + π
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SchrΓΆdinger Equation
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π βππ π₯, π‘
ππ‘= β
β2
2 π
π2π π₯, π‘
ππ₯2+ π π₯, π‘ π π₯, π‘
β’ Not valid for relativistic particles (relativistic particles require Dirac Equation)
β’ Not valid for photons (light requires quantization of the electromagnetic field, which is called Second Quantization)
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β’ What is really the wave function π π₯, π‘ or what is its physical meaning ?
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β’ Actually, we just know about the physical meaning of π π₯, π‘ 2.
β’ The SchrΓΆdinger Equation is for π π₯, π‘ .
β’ π π₯, π‘ 2 describes the probability density of finding the wave-particle at point π₯ at time π‘.
β’ However, we donβt have an equation for π π₯, π‘ 2.
A Few Remarks
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Probability Density and Probability
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Because π π₯, π‘ 2 describes the probability density, then:
π π₯, π‘ 2 ππ₯ gives us the probability of finding the wave-particle between (π₯) and (π₯ + ππ₯) at time (π‘).
π₯1π₯2 π π₯, π‘ 2ππ₯ gives us the probability of
finding the wave-particle between (π₯1) and (π₯2) at time (π‘).
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The probability of finding the wave-particle somewhere between ββ,βwill always be 1 at any time:
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ΰΆ±ββ
+β
π π₯, π‘ 2ππ₯ = 1
Normalization Condition
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Example 6.4
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π π₯, π‘ = π΄ πβπΌ π₯ π ΰ΅β π πΈ π‘β
π π₯, π‘ 2 = π΄ 2 πβ2πΌ π₯
1 = ΰΆ±ββ
+β
π π₯, π‘ 2ππ₯
= 2 π΄ 2ΰΆ±0
+β
πβ2 πΌ π₯ ππ₯ = 2 π΄ 2πβ2 πΌ π₯
β2 πΌπ₯=0
π₯=+β
=π΄ 2
πΌ
π΄ = πΌ ππ π
= ΰΆ±ββ
+β
π΄ 2πβ2πΌ π₯ ππ₯
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How can we get predictions from the wave function for physical quantities (position, energy, linear momentum,
angular momentum, β¦) ??
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6.2 Physical Observables and Expectation Values
We need to confront those predictions against experimental measurements !!
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Expectation Value for π₯ :
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πππ =π1 πππ1 + π2 πππ2 + π3 πππ3 + π4 πππ4 +β―
π1 + π2 +π3 + π4 +β―
=Οπ ππ ππππ
Οπ ππ
π₯ =ββ+β
π π₯ π₯ ππ₯
ββ+β
π π₯ ππ₯=ββ+β
π π₯, π‘ 2 π₯ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
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Expectation Value for π π₯ :
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π π₯ =ββ+β
π π₯, π‘ 2 π π₯ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
Expectation Value for π₯2:
π₯2 =ββ+β
π π₯, π‘ 2 π₯2ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
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Expectation Value for π:
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π =ββ+β
π π₯, π‘ 2 π ππ₯
ββ+β
π π₯, π‘ 2 ππ₯=ββ+β
πβ π₯, π‘ π π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
= π π π₯, π‘
π
ππ₯π π₯, π‘ = π π π π₯, π‘βπ β
βπ βπ
ππ₯π π₯, π‘ = β π π π₯, π‘
=ββ+β
πβ π₯, π‘ βπ βπππ₯
π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
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Operator for Linear Momentum:
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π =ββ+β
πβ π₯, π‘ βπ βπππ₯
π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
β π βπ
ππ₯β‘ ΰ·π
=ββ+β
πβ π₯, π‘ ΖΈπ π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
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Fall 2018 Prof. Sergio B. Mendes 19
π2
=ββ+β
πβ π₯, π‘ β β2π2
ππ₯2π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
Expectation Value for π2:
=ββ+β
πβ π₯, π‘ βπ βπππ₯
βπ βπππ₯
π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
=ββ+β
π π₯, π‘ 2 π2ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
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Expectation Value for πΈ:
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πΈ =ββ+β
π π₯, π‘ 2 πΈ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯=ββ+β
πβ π₯, π‘ πΈ π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
= πΈ π π₯, π‘
π
ππ‘π π₯, π‘ = β π π π π₯, π‘π β
π βπ
ππ‘π π₯, π‘ = βπ π π₯, π‘
=ββ+β
πβ π₯, π‘ π βπππ‘
π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
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Operator for Energy:
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πΈ =ββ+β
πβ π₯, π‘ π βπππ‘
π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
=ββ+β
πβ π₯, π‘ πΈ π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
π βπ
ππ‘β‘ πΈ
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Predicting Outcomes from the Theory:
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π π₯ =ββ+β
πβ π₯, π‘ π π₯ π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
π =ββ+β
πβ π₯, π‘ ΖΈπ π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯ΰ·π β‘ βπ β
π
ππ₯
πΈ =ββ+β
πβ π₯, π‘ πΈ π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯πΈ β‘ π β
π
ππ‘
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An useful result to rememberfrom Statistical Theory
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ππ₯2 β‘ π₯ β π₯ 2 = π₯2 β 2 π₯ π₯ + π₯ 2
= π₯2 β 2 π₯ π₯ + π₯ 2
= π₯2 β π₯ 2
= π₯2 β 2 π₯ π₯ + π₯ 2
= π₯2 β 2 π₯ 2 + π₯ 2
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SchrΓΆdinger Equation becomes simpler when the potential energy π
does not depend on time π‘
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Time Independent: π π₯, π‘ = π π₯
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π βππ π₯, π‘
ππ‘= β
β2
2 π
π2π π₯, π‘
ππ₯2+ π π₯, π‘π π₯, π‘π π₯
π π₯, π‘ β‘ π π₯ π π‘
= ββ2
2 ππ π‘
π2π π₯
ππ₯2+ π π₯ π π₯ π π‘
π β1
π π‘
ππ π‘
ππ‘= β
β2
2 π
1
π π₯
π2π π₯
ππ₯2+ π π₯ = πΈ
π β π π₯ππ π‘
ππ‘
1
π π₯ π π‘Γ
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Time Dependence of Wave Function
Fall 2018 Prof. Sergio B. Mendes 26
π π‘ = π ΰ΅βπ πΈ π‘β
π β1
π π‘
ππ π‘
ππ‘= πΈ
π π₯, π‘ β‘ π π₯ π π‘ = π π₯ π ΰ΅βπ πΈ π‘β
π π₯, π‘ 2 = π π₯ π ΰ΅βπ πΈ π‘β2= π π₯ 2
= π π₯ πβ π π π‘
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Fall 2018 Prof. Sergio B. Mendes 27
ββ2
2 π
1
π π₯
π2π π₯
ππ₯2+ π π₯ = πΈ
ββ2
2 π
π2π π₯
ππ₯2+ π π₯ π π₯ = πΈ π π₯
Time-Independent SchrΓΆdinger Equation:
π π₯ Γ
π π₯, π‘ = π π₯ π ΰ΅βπ πΈ π‘β
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Requirements on the Wave Function:
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β’ π π₯ and ππ π₯
ππ₯must be finite
everywhere.
β’ π π₯ and ππ π₯
ππ₯must be single-valued
everywhere.
β’ π π₯ and ππ π₯
ππ₯must be continuous (no
jumps) everywhere, at least where π π₯is finite.
β’ π π₯ β 0 when π₯ β Β±β .
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Predicting Expected Values for Physical Observables :
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π₯ =ββ+β
πβ π₯, π‘ π₯ π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯
π =ββ+β
πβ π₯, π‘ ΖΈπ π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯ΰ·π β‘ βπ β
π
ππ₯
πΈ =ββ+β
πβ π₯, π‘ πΈ π π₯, π‘ ππ₯
ββ+β
π π₯, π‘ 2 ππ₯πΈ β‘ π β
π
ππ‘
ΰ·π₯ β‘ π₯
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Fall 2018 Prof. Sergio B. Mendes 30
6.3 Infinite Square-Well Potential
ββ2
2 π
π2π π₯
ππ₯2= πΈ π π₯+ π π₯ π π₯
βπ2π π₯
ππ₯2=2 π πΈ
β2π π₯ = π2 π π₯
π π₯ = π΄ π ππ π π₯ + π΅ πππ π π₯
0 β€ π₯ β€ πΏ
π β‘2 π πΈ
β2
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Fall 2018 Prof. Sergio B. Mendes 31
Infinite Square-Well Potential, cont.
π π₯ = π΄ π ππ π π₯ + π΅ πππ π π₯ 0 β€ π₯ β€ πΏ
π π₯ = 0 π₯ β₯ πΏπ₯ β€ 0 and
π π₯ = 0 π π₯ = 0
π π₯ = π΄ π ππ π π₯ + π΅ πππ π π₯
π π₯ = 0 = π΅ = 0
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Fall 2018 Prof. Sergio B. Mendes 32
Infinite Square-Well Potential, cont.
π π₯ = π΄ π ππ π π₯ 0 β€ π₯ β€ πΏπ π₯ = 0 π π₯ = 0
π π₯ = π΄ π ππ π π₯
π π₯ = πΏ = π΄ π ππ π πΏ = 0
ππ = ππ
πΏ
ππ π₯ = π΄ π ππ ππ π₯
π = 1, 2, 3, β¦
= π΄ π ππ ππ
πΏπ₯
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Discrete Energy Levels:
Fall 2018 Prof. Sergio B. Mendes 33
=2 π πΈπβ2
ππ
πΏ= ππ
π = 1, 2, 3, β¦
ππ π₯ = π΄ π ππ ππ
πΏπ₯
πΈπ = π2β2 π2
2 π πΏ2
ππ π₯, π‘ 2 = π΄ 2 π ππ2 ππ
πΏπ₯
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Normalizing the Wave Function:
Fall 2018 Prof. Sergio B. Mendes 34
π π₯, π‘ 2 = ππ π₯, π‘ 2 = π΄ 2 π ππ2 ππ
πΏπ₯
ΰΆ±ββ
+β
π π₯, π‘ 2ππ₯ = 1
ΰΆ±0
πΏ
π΄ 2 π ππ2 ππ
πΏπ₯ ππ₯ = 1
π΄ 2πΏ
2= 1 π΄ =
2
πΏ
π ππ2 π =1
21 β πππ 2π
0 β€ π₯ β€ πΏ
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Bringing All Together:
Fall 2018 Prof. Sergio B. Mendes 35
π π₯, π‘ = ππ π₯ πβ π π π‘
= π΄ π ππ ππ π₯ πβ π π π‘
=π΄
2 ππ+π ππ π₯ β πβπ ππ π₯ πβ π π π‘
=π΄
2 ππ+π ππ π₯ β π π‘ β πβπ ππ π₯ + π π‘
π β‘πΈ
β
Two Counter-Propagating Waves
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A More Realistic Potential Energy:
Fall 2018 Prof. Sergio B. Mendes 36
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6.4 Finite Square-Well Potential
Fall 2018 Prof. Sergio B. Mendes 37
ββ2
2 π
π2π π₯
ππ₯2+ π π₯ π π₯ = πΈ π π₯
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Region II: 0 β€ π₯ β€ πΏ
Fall 2018 Prof. Sergio B. Mendes 38
ββ2
2 π
π2π π₯
ππ₯2= πΈ π π₯+ π π₯ π π₯
βπ2π π₯
ππ₯2=2 π πΈ
β2π π₯ = π2 π π₯
ππΌπΌ π₯ = π΄πΌπΌ π ππ π π₯ + π΅πΌπΌ πππ π π₯
π β‘2 π πΈ
β2
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Region I: π₯ β€ 0
Fall 2018 Prof. Sergio B. Mendes 39
ββ2
2 π
π2π π₯
ππ₯2+ π π₯ = πΈ π π₯π π₯
π2π π₯
ππ₯2=2π ππ β πΈ
β2π π₯ = πΌ2 π π₯
ππΌ π₯ = π΄πΌπβπΌ π₯ + π΅πΌ π
+πΌ π₯
πΌ β‘2 π ππ β πΈ
β2
ππ
= π΅πΌ π+πΌ π₯
π΄πΌ = 0to prevent the wave function to diverge when π₯ β ββ
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Region III: π₯ β₯ πΏ
Fall 2018 Prof. Sergio B. Mendes 40
ββ2
2 π
π2π π₯
ππ₯2+ π π₯ = πΈ π π₯π π₯
π2π π₯
ππ₯2=2π ππ β πΈ
β2π π₯ = πΌ2 π π₯
ππΌπΌπΌ π₯ = π΄πΌπΌπΌ πβπΌ π₯ + π΅πΌπΌπΌ π
+πΌ π₯
πΌ β‘2 π ππ β πΈ
β2
ππ
= π΄πΌπΌπΌ πβπΌ π₯
π΅πΌπΌπΌ = 0to prevent the wave function to diverge when π₯ β +β
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Bringing All the Pieces Together:
Fall 2018 Prof. Sergio B. Mendes 41
ππΌπΌ π₯ = π΄πΌπΌ π ππ π π₯ + π΅πΌπΌ πππ π π₯
ππΌπΌπΌ π₯ = π΄πΌπΌπΌ πβπΌ π₯
ππΌ π₯ = π΅πΌ π+πΌ π₯
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Boundary Conditions for Wave Function:
Fall 2018 Prof. Sergio B. Mendes 42
π₯ = 0 π΅πΌ = π΅πΌπΌ
π₯ = πΏ π΄πΌπΌ π ππ π πΏ + π΅πΌπΌ πππ π πΏ = π΄πΌπΌπΌ πβπΌ πΏ
ππΌπΌ π₯ = π΄πΌπΌ π ππ π π₯ + π΅πΌπΌ πππ π π₯
ππΌπΌπΌ π₯ = π΄πΌπΌπΌ πβπΌ π₯
ππΌ π₯ = π΅πΌ π+πΌ π₯
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Boundary Conditions for the Derivative:
Fall 2018 Prof. Sergio B. Mendes 43
πππΌπΌ π₯
ππ₯= π΄πΌπΌ π πππ π π₯ β π΅πΌπΌ π π ππ π π₯
πππΌπΌπΌ π₯
ππ₯= βπ΄πΌπΌπΌ πΌ π
βπΌ π₯
πππΌ π₯
ππ₯= π΅πΌ πΌ π
+πΌ π₯
π₯ = 0 π΅πΌ πΌ = π΄πΌπΌ π
π₯ = πΏ π΄πΌπΌ π πππ π πΏ β π΅πΌπΌ π π ππ π πΏ = βπ΄πΌπΌπΌ πΌ πβπΌ πΏ
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After Some Algebra (Appendix 1):
Fall 2018 Prof. Sergio B. Mendes 44
2 π‘ππβ1ππ β πΈ
πΈ+ π β 1 π =
2 π πΈ
β2πΏ
ππ β‘2π πΈπβ2
πΌπ β‘2π ππ β πΈπ
β2π = 1, 2, 3, β¦
ππΌ,π π₯ = π΅πΌπΌ,π π+πΌπ π₯
ππΌπΌ,π π₯ = π΅πΌπΌ,ππΌπππ
π ππ ππ π₯ + πππ ππ π₯
ππΌπΌπΌ,π π₯ = π΅πΌπΌ,π π+πΌπ πΏ
πΌπππ
π ππ ππ πΏ + πππ ππ πΏ πβπΌπ π₯
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Fall 2018 Prof. Sergio B. Mendes 45
π β‘2 π πΏ2 ππ
β22 π‘ππβ1
1
π₯β 1 + π β 1 π = π π₯ π₯ β‘
πΈ
ππ
π = 10
πΈ1ππ
πΈ2ππ
πΈ3ππ
πΈ4ππ
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Discrete Energy Levelsand Associated Wave Functions
Fall 2018 Prof. Sergio B. Mendes 46
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PhET
Fall 2018 Prof. Sergio B. Mendes 47
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Number of Confined Wave Functions
Fall 2018 Prof. Sergio B. Mendes 48
ππ’ππππ ππ πΆπππππππ πππ£π πΉπ’πππ‘ππππ = πΌππ‘πππππ
π+ 1
= πΌππ‘ππππ2 π πΏ2 ππβ2 π2
+ 1
= πΌππ‘ππππ8 π πΏ2 ππ
β2+ 1
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Classical Physics
Fall 2018 Prof. Sergio B. Mendes 49
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6.7 Potential Barrier
Fall 2018 Prof. Sergio B. Mendes 50
πΈ β₯ ππ
Total energy higher than the barrier
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Region I: π₯ β€ 0
Fall 2018 Prof. Sergio B. Mendes 51
ββ2
2 π
π2π π₯
ππ₯2= πΈ π π₯+ π π₯ π π₯
βπ2π π₯
ππ₯2=2 π πΈ
β2π π₯ = π2 π π₯
ππΌ π₯ = π΄πΌ π+π π π₯ + π΅πΌ π
βπ π π₯
π β‘2 π πΈ
β2
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Region III: π₯ β₯ πΏ
Fall 2018 Prof. Sergio B. Mendes 52
ββ2
2 π
π2π π₯
ππ₯2= πΈ π π₯+ π π₯ π π₯
βπ2π π₯
ππ₯2=2 π πΈ
β2π π₯ = π2 π π₯
ππΌπΌπΌ π₯ = π΄πΌπΌπΌ π+π π π₯ + π΅πΌπΌπΌ π
βπ π π₯
π β‘2 π πΈ
β2
= π΄πΌπΌπΌ π+π π π₯ π΅πΌπΌπΌ = 0
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Region II: 0 β€ π₯ β€ πΏ
Fall 2018 Prof. Sergio B. Mendes 53
ββ2
2 π
π2π π₯
ππ₯2+ π π₯ = πΈ π π₯π π₯
βπ2π π₯
ππ₯2=2 π πΈ β ππ
β2π π₯ = πβ²2 π π₯
πβ² β‘2 π πΈ β ππ
β2
ππΌπΌ π₯ = π΄πΌπΌ π+π πβ² π₯ + π΅πΌπΌ π
βπ πβ² π₯
ππ
![Page 54: CHAPTER 6 300 fall 19...Β Β· 2019-10-21Β Β· Fall 2018 Prof. Sergio B. Mendes 9 πβ , =β β2 2 2 , 2 + , , β’ Not valid for relativistic particles (relativistic particles require](https://reader034.fdocuments.net/reader034/viewer/2022042112/5e8dac9f6373cf1f735d1c80/html5/thumbnails/54.jpg)
Bringing All the Pieces Together:
Fall 2018 Prof. Sergio B. Mendes 54
ππΌ π₯ = π΄πΌ π+π π π₯ + π΅πΌ π
βπ π π₯
ππΌπΌπΌ π₯ = π΄πΌπΌπΌ π+π π π₯
ππΌπΌ π₯ = π΄πΌπΌ π+π πβ² π₯ + π΅πΌπΌ π
βπ πβ² π₯
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Boundary Conditions for Wave Function:
Fall 2018 Prof. Sergio B. Mendes 55
ππΌ π₯ = π΄πΌ π+π π π₯ + π΅πΌ π
βπ π π₯
ππΌπΌπΌ π₯ = π΄πΌπΌπΌ π+π π π₯
ππΌπΌ π₯ = π΄πΌπΌ π+π πβ² π₯ + π΅πΌπΌ π
βπ πβ² π₯
π₯ = 0
π₯ = πΏ
π΄πΌ + π΅πΌ = π΄πΌπΌ + π΅πΌπΌ
π΄πΌπΌ π+π πβ² πΏ + π΅πΌπΌ π
βπ πβ² πΏ = π΄πΌπΌπΌ π+π π πΏ
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Boundary Conditions for the Derivative:
Fall 2018 Prof. Sergio B. Mendes 56
πππΌπΌ π₯
ππ₯= π πβ² π΄πΌπΌ π
+π πβ² π₯ β π΅πΌπΌ πβπ πβ² π₯
πππΌπΌπΌ π₯
ππ₯= π π π΄πΌπΌπΌ π
+π π π₯
πππΌ π₯
ππ₯= π π π΄πΌ π
+π π π₯ β π΅πΌ πβπ π π₯
π₯ = 0
π₯ = πΏ
π π΄πΌ β π΅πΌ = πβ² π΄πΌπΌ β π΅πΌπΌ
πβ² π΄πΌπΌ π+π πβ² πΏ β π΅πΌπΌ π
βπ πβ² πΏ = π π΄πΌπΌπΌ π+π π πΏ
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Reflectance and Transmittance
Fall 2018 Prof. Sergio B. Mendes 57
π β‘π΄πΌπΌπΌ
2
π΄πΌ2
π β‘π΅πΌ
2
π΄πΌ2
ππΌ π₯= π΄πΌ π
+π π π₯
+ π΅πΌ πβπ π π₯
ππΌπΌπΌ π₯= π΄πΌπΌπΌ π
+π π π₯
π = 1 +π0
2 π ππ2 πβ²πΏ
4 πΈ πΈ β π0
β1
πβ²πΏ = π π π = 1
see Appendix 2
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Fall 2018 Prof. Sergio B. Mendes 58
πΈ β€ ππPotential Barrier:
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Region II: π₯ β€ 0
Fall 2018 Prof. Sergio B. Mendes 59
βπ2π π₯
ππ₯2=2 π πΈ β ππ
β2π π₯ = β π 2 π π₯
π β‘2 π ππ β πΈ
β2
ππΌπΌ π₯ = π΄πΌπΌ π+ π π₯ + π΅πΌπΌ π
β π π₯
ββ2
2 π
π2π π₯
ππ₯2+ π π₯ = πΈ π π₯π π₯ππ
= π πβ²
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Tunneling Probability
Fall 2018 Prof. Sergio B. Mendes 60
π = 1 +π0
2 π ππβ2 π πΏ
4 πΈ π0 β πΈ
β1
πΈ
π0
β 16 πΈ π0 β πΈ
π02 πβ 2 π πΏ
π β‘2 π ππ β πΈ
β2
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Tunneling in Classical Optics
Fall 2018 Prof. Sergio B. Mendes 61
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Quantum Tunneling
Fall 2018 Prof. Sergio B. Mendes 62
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PhET
Fall 2018 Prof. Sergio B. Mendes 63
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Scanning Tunneling Microscope
Fall 2018 Prof. Sergio B. Mendes 64
Heinrich Rohrer (right, 1933β ) and GerdBinnig (1947β ) received the Nobel Prize for Physics in 1986 for their design of the scanning tunneling micro-scope.
The Swiss Rohrer was educated at the Swiss Federal Institute of Technology in Zurich and joined the IBM Research Laboratory in Zurich in 1963.
The German Binnig received his doctorate from the University of Frankfurt (Germany) in 1978 and then joined the same IBM Research Laboratory. He moved to the IBM Physics Group in Munich in 1984
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Fall 2018 Prof. Sergio B. Mendes 65
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Xe Atoms on Ni Surface
Fall 2018 Prof. Sergio B. Mendes 66
Photo taken with an STM, show xenon atoms placed on a nickel surface. The xenon atoms are 0.16 nm high and adjacent xenon atoms are 0.5 nm apart (the vertical scale has been exaggerated). The small force between the STM tip and an atom is enough to drag one xenon atom at a time across the nickel. The nickel atoms are represented by the black-and-white stripes on the horizontal surface. The image is magnified about 5 million times.
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Ammonia Inversion
Fall 2018 Prof. Sergio B. Mendes 67
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Field Effect Transistor
Fall 2018 Prof. Sergio B. Mendes 68
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6.6 Harmonic Oscillator
Fall 2018 Prof. Sergio B. Mendes 69
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Potential Energy and Wave Function
Fall 2018 Prof. Sergio B. Mendes 70
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Lowest Order Solutions:
Fall 2018 Prof. Sergio B. Mendes 71
πΈπ = π +1
2β π
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N = 10
Fall 2018 Prof. Sergio B. Mendes 72
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6.5 Three-Dimensional, Time-Independent, SchrΓΆdinger Equation
Fall 2018 Prof. Sergio B. Mendes 73
ββ2
2 π
π2π π₯
ππ₯2+ π π₯ π π₯ = πΈ π π₯
ββ2
2 π
π2
ππ₯2+
π2
ππ¦2+
π2
ππ§2π π₯, π¦, π§ + π π₯, π¦, π§ π π₯, π¦, π§ = πΈ π π₯, π¦, π§
1D
3D
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Appendix 1
Fall 2018 Prof. Sergio B. Mendes 74
Matching Boundary Conditions on Finite Square-Well Potential
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Bringing All the Pieces Together:
Fall 2018 Prof. Sergio B. Mendes 75
ππΌπΌ π₯ = π΄πΌπΌ π ππ π π₯ + π΅πΌπΌ πππ π π₯
ππΌπΌπΌ π₯ = π΄πΌπΌπΌ πβπΌ π₯
ππΌ π₯ = π΅πΌ π+πΌ π₯
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Matching Boundary Conditions:
Fall 2018 Prof. Sergio B. Mendes 76
π₯ = 0 π΅πΌ = π΅πΌπΌ
π₯ = πΏ π΄πΌπΌ π ππ π πΏ + π΅πΌπΌ πππ π πΏ = π΄πΌπΌπΌ πβπΌ πΏ
ππΌπΌ π₯ = π΄πΌπΌ π ππ π π₯ + π΅πΌπΌ πππ π π₯
ππΌπΌπΌ π₯ = π΄πΌπΌπΌ πβπΌ π₯
ππΌ π₯ = π΅πΌ π+πΌ π₯
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Bound. Cond. for the Derivatives:
Fall 2018 Prof. Sergio B. Mendes 77
πππΌπΌ π₯
ππ₯= π΄πΌπΌ π πππ π π₯ β π΅πΌπΌ π π ππ π π₯
πππΌπΌπΌ π₯
ππ₯= βπ΄πΌπΌπΌ πΌ π
βπΌ π₯
πππΌ π₯
ππ₯= π΅πΌ πΌ π
+πΌ π₯
π₯ = 0 π΅πΌ πΌ = π΄πΌπΌ π
π₯ = πΏ π΄πΌπΌ π πππ π πΏ β π΅πΌπΌ π π ππ π πΏ = βπ΄πΌπΌπΌ πΌ πβπΌ πΏ
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Fall 2018 Prof. Sergio B. Mendes 78
π΅πΌ = π΅πΌπΌ
π΄πΌπΌ π ππ π πΏ + π΅πΌπΌ πππ π πΏ = π΄πΌπΌπΌ πβπΌ πΏ
π΅πΌ πΌ = π΄πΌπΌ π
π΄πΌπΌ π πππ π πΏ β π΅πΌπΌ π π ππ π πΏ = βπ΄πΌπΌπΌ πΌ πβπΌ πΏ
π΅πΌπΌ πΌ = π΄πΌπΌ π
π΄πΌπΌ π πππ π πΏ β π΅πΌπΌ π π ππ π πΏ = β πΌ π΄πΌπΌ π ππ π πΏ + π΅πΌπΌ πππ π πΏ
1
2
3
4
1 3&
2 4&
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Fall 2018 Prof. Sergio B. Mendes 79
π΅πΌπΌ πΌ = π΄πΌπΌ π
π΄πΌπΌ π πππ π πΏ β π΅πΌπΌ π π ππ π πΏ = β πΌ π΄πΌπΌ π ππ π πΏ + π΅πΌπΌ πππ π πΏ
π΅πΌπΌ πΌ π πππ π πΏ β π΅πΌπΌ π2 π ππ π πΏ = β πΌ π΅πΌπΌ πΌ π ππ π πΏ + π΅πΌπΌ π πππ π πΏ
πΌ π πππ π πΏ β π2 π ππ π πΏ = βπΌ2 π ππ π πΏ β πΌ π πππ π πΏ
2 πΌ π πππ π πΏ = π2 β πΌ2 π ππ π πΏ
2 πΌ π
π2 β πΌ2= π‘ππ π πΏ
π‘ππ 2 πΏ =2 π‘ππ πΏ
1 β π‘ππ2 πΏ= π‘ππ π πΏ
π‘ππ πΏ β‘πΌ
π
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Fall 2018 Prof. Sergio B. Mendes 80
π‘ππ 2 πΏ = π‘ππ π πΏ
2 πΏ + π β 1 π = π πΏ
2 π‘ππβ1πΌ
π+ π β 1 π = π πΏ
2 π‘ππβ1ππ β πΈ
πΈ+ π β 1 π =
2 π πΈ
β2πΏ
π = 1, 2, 3, β¦
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Fall 2018 Prof. Sergio B. Mendes 81
π β‘2 π πΏ2 ππ
β22 π‘ππβ1
1
π₯β 1 + π β 1 π = π π₯ π₯ β‘
πΈ
ππ
π = 10
πΈ1ππ
πΈ2ππ
πΈ3ππ
πΈ4ππ
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Number of Confined Wave Functions
Fall 2018 Prof. Sergio B. Mendes 82
ππ’ππππ ππ πΆπππππππ πππ£π πΉπ’πππ‘ππππ = πΌππ‘πππππ
π+ 1
= πΌππ‘ππππ2 π πΏ2 ππβ2 π2
+ 1
= πΌππ‘ππππ8 π πΏ2 ππ
β2+ 1
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Region I:
Fall 2018 Prof. Sergio B. Mendes 83
ππΌ,π π₯ = π΅πΌ,π π+πΌπ π₯
π΅πΌ = π΅πΌπΌ
= π΅πΌπΌ,π π+πΌπ π₯
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Region II:
Fall 2018 Prof. Sergio B. Mendes 84
ππΌπΌ,π π₯ = π΄πΌπΌ,π π ππ ππ π₯ + π΅πΌπΌ,π πππ ππ π₯
= π΅πΌπΌ,ππΌπππ
π ππ ππ π₯ + πππ ππ π₯
π΅πΌ = π΅πΌπΌ
π΅πΌ πΌ = π΄πΌπΌ π
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Region III:
Fall 2018 Prof. Sergio B. Mendes 85
ππΌπΌπΌ,π π₯ = π΄πΌπΌπΌ,π πβπΌπ π₯
= π΅πΌπΌ,π π+πΌπ πΏ
πΌπππ
π ππ ππ πΏ + πππ ππ πΏ πβπΌπ π₯
π΅πΌ = π΅πΌπΌ
π΄πΌπΌ π ππ π πΏ + π΅πΌπΌ πππ π πΏ = π΄πΌπΌπΌ πβπΌ πΏ
π΅πΌ πΌ = π΄πΌπΌ π
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All Regions:
Fall 2018 Prof. Sergio B. Mendes 86
ππΌ,π π₯ = π΅πΌπΌ,π π+πΌπ π₯
ππΌπΌ,π π₯ = π΅πΌπΌ,ππΌπππ
π ππ ππ π₯ + πππ ππ π₯
ππΌπΌπΌ,π π₯ = π΅πΌπΌ,π π+πΌπ πΏ
πΌπππ
π ππ ππ πΏ + πππ ππ πΏ πβπΌπ π₯
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Verification:
Fall 2018 Prof. Sergio B. Mendes 87
ππ π΅πΌπΌ,ππΌπππ
πππ ππ πΏ β π ππ ππ πΏ = βπΌπ π΅πΌπΌ,π π+πΌπ πΏ
πΌπππ
π ππ ππ πΏ + πππ ππ πΏ πβπΌπ πΏ
πππΌπππ
πππ ππ πΏ β π ππ ππ πΏ = βπΌππΌπππ
π ππ ππ πΏ + πππ ππ πΏ
πΌππΌπππ
β ππ π ππ ππ πΏ = βπππΌπππ
β πΌπ πππ ππ πΏ
π‘ππ ππ πΏ =βππ
πΌπππ
β πΌπ
πΌππΌπππ
β ππ
=2 ππ πΌπ
ππ2 β πΌπ
2
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Appendix 2
Fall 2018 Prof. Sergio B. Mendes 88
Matching Boundary Conditions on Barrier Potential
πΈ β₯ ππ
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Fall 2018 Prof. Sergio B. Mendes 89
π΄πΌ + π΅πΌ = π΄πΌπΌ + π΅πΌπΌ
π΄πΌπΌ π+π πβ² πΏ + π΅πΌπΌ π
βπ πβ² πΏ = π΄πΌπΌπΌ π+π π πΏ
π π΄πΌ β π΅πΌ = πβ² π΄πΌπΌ β π΅πΌπΌ
πβ² π΄πΌπΌ π+π πβ² πΏ β π΅πΌπΌ π
βπ πβ² πΏ = π π΄πΌπΌπΌ π+π π πΏ
πβ² π΄πΌπΌ π+π πβ² πΏ β π΅πΌπΌ π
βπ πβ² πΏ = π π΄πΌπΌ π+π πβ² πΏ + π΅πΌπΌ π
βπ πβ² πΏ
π 2 π΄πΌ β π΄πΌπΌ β π΅πΌπΌ = πβ² π΄πΌπΌ β π΅πΌπΌ
1
2
3
4
1 3
2 4
&
&
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Fall 2018 Prof. Sergio B. Mendes 90
πβ² π΄πΌπΌ π+π πβ² πΏ β π΅πΌπΌ π
βπ πβ² πΏ = π π΄πΌπΌ π+π πβ² πΏ + π΅πΌπΌ π
βπ πβ² πΏ
π΄πΌ =π π΄πΌπΌ + π΅πΌπΌ + πβ² π΄πΌπΌ β π΅πΌπΌ
2 π=π΄πΌπΌ π + πβ² + π΅πΌπΌ π β πβ²
2 π
π΄πΌπΌ π+π πβ² πΏ πβ² β π = π΅πΌπΌ π
βπ πβ² πΏ πβ² + π
π΅πΌπΌ = π΄πΌπΌ ππ 2 πβ²πΏ
πβ² β π
πβ² + π
π 2 π΄πΌ β π΄πΌπΌ β π΅πΌπΌ = πβ² π΄πΌπΌ β π΅πΌπΌ
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Fall 2018 Prof. Sergio B. Mendes 91
π΄πΌ =π΄πΌπΌ π + πβ² + π΅πΌπΌ π β πβ²
2 π= π΄πΌπΌ
πβ² + π 2 β ππ 2 πβ²πΏ πβ² β π 2
2 π πβ² + π
π΄πΌπΌπΌ = πβ π π πΏ π΄πΌπΌ π+π πβ² πΏ + π΅πΌπΌ π
βπ πβ² πΏ
= π΄πΌπΌ πβ π π πΏ π+π π
β² πΏ + ππ 2 πβ²πΏ
πβ² β π
πβ² + ππβπ π
β² πΏ
= π΄πΌπΌ πβ π π πΏ π+π π
β² πΏ2 πβ²
πβ² + π
π΅πΌπΌ = π΄πΌπΌ ππ 2 πβ²πΏ
πβ² β π
πβ² + π
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Fall 2018 Prof. Sergio B. Mendes 92
=π΄πΌπΌ π
π πβ²πΏ
πβ² + π
πβπ πβ²πΏ πβ² + π 2 β ππ π
β²πΏ πβ² β π 2
2 π
π΄πΌ = π΄πΌπΌπβ² + π 2 β ππ 2 π
β²πΏ πβ² β π 2
2 π πβ² + π
π΄πΌπΌπΌ =π΄πΌπΌ π
β π π πΏ π+π πβ² πΏ
πβ² + π2 πβ²
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Fall 2018 Prof. Sergio B. Mendes 93
π =π΄πΌπΌπΌ
2
π΄πΌ2
π΄πΌ =π΄πΌπΌ π
π πβ²πΏ
πβ² + π
πβπ πβ²πΏ πβ² + π 2 β ππ π
β²πΏ πβ² β π 2
2 π
π΄πΌπΌπΌ =π΄πΌπΌ π
β π π πΏ π+π πβ² πΏ
πβ² + π2 πβ²
π =π΄πΌπΌπΌ
2
π΄πΌ2=
16 πβ²2π2
πβ² + π 4 + πβ² β π 4 β 2 πβ² + π 2 πβ² β π 2 πππ 2 πβ²πΏ
![Page 94: CHAPTER 6 300 fall 19...Β Β· 2019-10-21Β Β· Fall 2018 Prof. Sergio B. Mendes 9 πβ , =β β2 2 2 , 2 + , , β’ Not valid for relativistic particles (relativistic particles require](https://reader034.fdocuments.net/reader034/viewer/2022042112/5e8dac9f6373cf1f735d1c80/html5/thumbnails/94.jpg)
Fall 2018 Prof. Sergio B. Mendes 94
π =π΄πΌπΌπΌ
2
π΄πΌ2=
16 πβ²2π2
πβ² + π 4 + πβ² β π 4 β 2 πβ² + π 2 πβ² β π 2 πππ 2 πβ²πΏ
=16 πβ²
2π2
πβ² + π 4 + πβ² β π 4 β 2 πβ² + π 2 πβ² β π 2 1 β 2 π ππ2 πβ²πΏ
1
π=
πβ² + π 4 + πβ² β π 4 β 2 πβ² + π 2 πβ² β π 2 +4 πβ² + π 2 πβ² β π 2π ππ2 πβ²πΏ
16 πβ²2 π2
=πβ² + π 2 β πβ² β π 2 2 +4 πβ² + π 2 πβ² β π 2π ππ2 πβ²πΏ
16 πβ²2 π2
= 1 +πβ² + π 2 πβ² β π 2 π ππ2 πβ²πΏ
4 πβ²2 π2
![Page 95: CHAPTER 6 300 fall 19...Β Β· 2019-10-21Β Β· Fall 2018 Prof. Sergio B. Mendes 9 πβ , =β β2 2 2 , 2 + , , β’ Not valid for relativistic particles (relativistic particles require](https://reader034.fdocuments.net/reader034/viewer/2022042112/5e8dac9f6373cf1f735d1c80/html5/thumbnails/95.jpg)
Fall 2018 Prof. Sergio B. Mendes 95
πβ² β‘2 π πΈ β ππ
β2
1
π= 1 +
πβ² + π 2 πβ² β π 2 π ππ2 πβ²πΏ
4 πβ²2 π2
π β‘2 π πΈ
β2
1
π= 1 +
πΈ β ππ β πΈ 2 π ππ2 πβ²πΏ
4 πΈ β ππ2 πΈ2
= 1 +ππ
2 π ππ2 πβ²πΏ
4 πΈ β ππ πΈ
= 1 +πβ² 2 β π 2 2 π ππ2 πβ²πΏ
4 πβ²2 π2
= 1 +π ππ2 πβ²πΏ
4πΈππβ 1
πΈππ
![Page 96: CHAPTER 6 300 fall 19...Β Β· 2019-10-21Β Β· Fall 2018 Prof. Sergio B. Mendes 9 πβ , =β β2 2 2 , 2 + , , β’ Not valid for relativistic particles (relativistic particles require](https://reader034.fdocuments.net/reader034/viewer/2022042112/5e8dac9f6373cf1f735d1c80/html5/thumbnails/96.jpg)
Appendix 3
Fall 2018 Prof. Sergio B. Mendes 96
Matching Boundary Conditions on Barrier Potential
πΈ β€ ππ
![Page 97: CHAPTER 6 300 fall 19...Β Β· 2019-10-21Β Β· Fall 2018 Prof. Sergio B. Mendes 9 πβ , =β β2 2 2 , 2 + , , β’ Not valid for relativistic particles (relativistic particles require](https://reader034.fdocuments.net/reader034/viewer/2022042112/5e8dac9f6373cf1f735d1c80/html5/thumbnails/97.jpg)
Fall 2018 Prof. Sergio B. Mendes 97
π β‘ π πβ²πβ² β‘2 π πΈ β ππ
β2π β‘
2 π ππ β πΈ
β2
π΄πΌ =π΄πΌπΌ π
π πβ²πΏ
πβ² + π
πβπ πβ²πΏ πβ² + π 2 β ππ π
β²πΏ πβ² β π 2
2 π
π΄πΌπΌπΌ =π΄πΌπΌ π
β π π πΏ π+π πβ² πΏ
πβ² + π2 πβ²
![Page 98: CHAPTER 6 300 fall 19...Β Β· 2019-10-21Β Β· Fall 2018 Prof. Sergio B. Mendes 9 πβ , =β β2 2 2 , 2 + , , β’ Not valid for relativistic particles (relativistic particles require](https://reader034.fdocuments.net/reader034/viewer/2022042112/5e8dac9f6373cf1f735d1c80/html5/thumbnails/98.jpg)
Fall 2018 Prof. Sergio B. Mendes 98
π =π΄πΌπΌπΌ
2
π΄πΌ2