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INE: Revision
INTRODUCTION TO NANOELECTRONICS Week-5, Lecture-2
Sneh Saurabh 30th January, 2019
Introduction to Nanoelectronics: S. Saurabh Overview 2
Particle in a Box : Time-Independent Solution
퐿 = 푛휋 ℏ
풏ퟐ흅ퟐℏퟐ ⇒ 푬 = Energy is quantized, If 푳 is small 푬 is large 풏 ퟐ풎푳ퟐ 풏 Quantum Mechanics 휓 푥 = 퐴 sin
When we normalize: 퐴 = (Problem: 5)
휓 = sin Successive states are even/odd functions
Particle in a Box The stationary states are:
풏ퟐ흅ퟐℏퟐ /ℏ Ψ (푥, 푡)= sin 푒 ퟐ풎푳ퟐ
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Particle in a Box : Wave functions Particle in a Box : General Time-Dependent Solution
The general solution is:
풏ퟐ흅ퟐℏퟐ /ℏ 휳 풙, 풕 = ∑ 풄 sin 푒 ퟐ풎푳ퟐ [Check the solution by putting it in Schrödinger 풏 ퟏ 풏 equation]
Given initial condition 휳 풙, 풕 , 풄풏 can be computed: 2 푛휋푥 휳 풙, ퟎ = 풄 sin 풏 퐿 퐿 풏 ퟏ
⇒ 풄 = ∫ sin 휳 풙, ퟎ 푑푥 [Apply formula of Fourier series] 풏
. For the given initial condition, once 풄풏 is computed, the wave function 휳 풙, 풕 is fully known . Once the wave function is fully known, all the dynamical variables can be computed for the A particle in a box: Why E and F look weird? quantum mechanical system
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Particle in a Box : Problem (2)
Problem 12:
Find the probability that a particle trapped in a box of length 퐿 can be found between 0.45퐿 and 0.55퐿 for: a. the ground state Quantum Mechanics b. the first excited state
Answer: Particle in a Finite a. 19% Potential Well b. 0.65% Source: Arthur Beiser, "Concepts of Modern Physics"
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Particle in a Finite Potential Well: Problem Statement Particle in a Finite Potential Well: Schrödinger Equation
. Classically, the particle bounces back-forth on Outside the well (Region I and Region III): striking at the edge of Region II and never enters ℏퟐ 풅ퟐ흍 Region I or Region III +(푬 − 푈 )흍 = ퟎ ퟐ풎 풅풙ퟐ 풑
. In quantum mechanics, there is a finite probability that the particle will enter Region I and Region III on 푬풑 Let 훼 = striking at the edges of Region II ℏ 휶풙 휶풙 휶풙 휶풙 흍푰 = 푪풆 + 푫풆 and 흍푰푰푰 = 푬풆 + 푭풆
. Finite Potential: realistic Within the well (Region II) (similar Applying boundary condition: 휓 →0 as 푥 → ±∞: to particle in a box): 휶풙 휶풙 . Potential energy 푉(푥): 흍푰 = 푪풆 and 흍푰푰푰 = 푭풆 ퟐ ퟐ ℏ 풅 흍 푉(푥) = 0 for − ≤푥≤+ ퟐ + 푬풑흍 = ퟎ Need to find A, B, C and F based on boundary condition: ퟐ풎 풅풙 푉(푥) = 푈 else where 푬 Let β= 풑 (흍푰)풙 푳/ퟐ =(흍푰푰)풙 푳/ퟐ and (흍푰푰푰 )풙 푳/ퟐ =(흍푰푰)풙 푳/ퟐ ℏ . Energy of particle 푬풑 is less than (풅흍푰/풅풙)풙 푳/ퟐ =(풅흍푰푰/풅풙)풙 푳/ퟐ and (풅흍푰푰푰/풅풙)풙 푳/ퟐ = the potential of well 푈 흍푰푰 = 푨 sin 훽푥 + 푩 cos 훽푥 (풅흍푰푰/풅풙)풙 푳/ퟐ
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Particle in a Symmetric Potential Well: Parity Particle in a Finite Potential Well: Possible Solutions
. The potential well has a line of symmetry down the center of the box Within the well (Region II) Within the well (Region II) (푥 = 0) Odd Solution: considering Even Parity Solution:
. Basic considerations of symmetry demand that the probability for 휓 = 퐴 sin 훽푥 finding the particle at −푥 be the same as that at +푥 휓 = 퐵푐표푠훽푥 Outside the well (Region I Outside the well (Region I and Ψ 푥 = Ψ −푥 , and Region III): Same as with Region III): Even Parity Solution ∗ ∗ ⇒ 휳 (풙)휳 풙 = 휳 (−풙)휳 −풙 휓 = 퐶푒 and 휓 = 퐹푒
. This condition is satisfied if the Ψ 푥 has an either an even parity or Even Parity: odd parity Applying boundary condition (Even function): Applying boundary Ψ 푥 =Ψ −푥 condition (Odd function): . The condition is violated if the Ψ 푥 is a linear combination of even (휓 ) / =(휓 ) / and (푑휓 /푑푥) / =(푑휓 /푑푥) / and odd parity functions 퐹푒 / = 퐴 sin 훽퐿/2 and Odd Parity: 퐹푒 / = 퐵 cos 훽퐿/2 and −퐹훼푒 / =−퐵훽 sin 훽퐿/2 − 퐹훼푒 / = 퐴훽 cos 훽퐿/2 Ψ 푥 = −Ψ −푥 휶 = 휷 퐭퐚퐧 휷푳/ퟐ 휶 =−휷 퐜퐨퐭 휷푳/ퟐ
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Particle in a Finite Potential Well: Graphical Solution Particle in a Finite Potential Well: Nature of solution (1)
Transform these equations by substitution and elimination in a 푧 − 푧 = 푧 tan 푧 form that is easy to solve graphically (so that we can get some intuitive understanding of the solution): ⇒ − ퟏ = tan 푧 [Even Solutions] 푬풑 푬 We had assumed: 훼 = and β= 풑 ℏ ℏ
푬풑 . 푧 = = Let us assume: 푢 = , 푧 = and 푧 = ℏ 휶 = 휷 tan 휷푳/ퟐ or ℏ 휶 =−휷 cot 휷푳/ퟐ ⇒ 푧 represents energy of Then following relationship holds: particle
. Both 휶 and 휷 depends on energy, and only some 푧 − 푧 = 푢 and = = . Energy is quantized, finite number of solutions for finite particular value of 휶 and 휷 potential well . 푧 = yield a valid solution ℏ ⇒ Only a certain value 휶 = 휷 tan 휷푳/ퟐ ≡ 휶 =−휷 cot 휷푳/ퟐ ≡ . Potential height increases, number of possible solutions ⇒ 푧 represents potential increase of energy 푬풑 will be barrier valid 푧 − 푧 = 푧 tan 푧 푧 − 푧 =−푧 cot 푧 . Similar argument holds for odd solutions
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Particle in a Finite Potential Well: Nature of solution (2) Particle in a Finite Potential Well: Nature of solution (3)
. When potential height increases 푈 →∞, 푧 →∞ Infinite solution possible Energy levels for an The solutions tend to be: electron in a finite 풏흅 풛 = potential well ퟐ (푈 = 64 푒푉, 퐿 = 푬 풏흅 풑 = 0.39 푛푚) and infinite ℏ ퟐ potential well (퐿 = 풏ퟐ흅ퟐℏퟐ 푬 = 0.39 푛푚) 풑 ퟐ풎푳ퟐ . 푧 represents energy of particle Solutions are same as in case of particle in . 푧 represents potential barrier a box Source: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/pbox.html
. When potential height decreases 푈 →0, . For a finite potential well, wave 풏ퟐ흅ퟐℏퟐ . 푬 = 푧 →0 functions extend across the well 풑 ퟐ풎푳ퟐ One solution possible . Effective length of potential well . Energy levels for a finite potential well are At least one bound state always possible increases lower compared to infinite potential well for a given 푈
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References
David J. Griffiths, “Introduction to Quantum Mechanics”
Feynman, Leighton, Sands, “The Feynman Lectures on Physics”, Vol. 3
Arthur Beiser, "Concepts of Modern Physics”
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