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Lecture 12: Particle in 1D Boxes, Simple Harmonic Oscillators Lecture 12: Particle in 1D boxes, Simple Harmonic Oscillators U(x) U→∞→∞→∞ ψ(x) U→∞→∞→∞ n=2 n=0 n=1 n=3 x Lecture 12, p 1 This week and last week are critical for the course: Week 3, Lectures 7-9: Week 4, Lectures 10-12: Light as Particles Schrödinger Equation Particles as waves Particles in infinite wells, finite wells Probability Uncertainty Principle Midterm Exam Monday, week 5 It will cover lectures 1-11 and some aspects of lecture 12 (not SHOs). Practice exams: Old exams are linked from the course web page. Review Sunday before Midterm Office hours: Sunday and Monday Next week: Homework 4 covers material in lecture 10 – due on Thur. after midterm. We strongly encourage you to look at the homework before the midterm! Discussion : Covers material in lectures 10-12. There will be a quiz . Lab: Go to 257 Loomis (a computer room). You can save a lot of time by reading the lab ahead of time – It’s a tutorial on how to draw wave functions. Lecture 12, p 2 PropertiesProperties ofof BoundBound StatesStates Several trends exhibited by the particle-in-box states are generic to bound state wave functions in any 1D potential (even complicated ones). 1: The overall curvature of the wave function increases with increasing kinetic energy. ℏ2d 2ψ ( x ) p 2 − = for a sine wave 2m dx2 2 m ψ(x) 2: The lowest energy bound state always has finite kinetic n=2 n=1 n=3 energy -- called “zero-point” energy. Even the lowest energy bound state requires some wave function curvature (kinetic energy) to satisfy boundary conditions. 0 L x 3: The n th wave function (eigenstate) has (n-1) zero-crossings. Larger n means larger E (and p), which means more wiggles. 4: If the potential U(x) has a center of symmetry (such as the center of the well above), the eigenstates will be, alternately, even and odd functions about that center of symmetry. Lecture 12, p 3 Act 1 The wave function below describes a quantum particle in a range ∆x: ψ( x) 1. In what energy level is the particle? n = x (a) 7 (b) 8 (c) 9 ∆x 2. What is the approximate shape of the potential U(x) in which this particle is confined? U(x) (a) U(x) (b) U(x) (c) E E E ∆x ∆x ∆xLecture 12, p 4 Solution The wave function below describes a quantum particle in a range ∆x: ψ( x) 1. In what energy level is the particle? n = x (a) 7 (b) 8 (c) 9 Eight nodes. ∆x Don’t count the boundary conditions. 2. What is the approximate shape of the potential U(x) in which this particle is confined? U(x) (a) U(x) (b) U(x) (c) E E E ∆x ∆x ∆xLecture 12, p 5 Solution The wave function below describes a quantum particle in a range ∆x: ψ( x) 1. In what energy level is the particle? n = x (a) 7 (b) 8 (c) 9 Eight nodes. ∆x Don’t count the boundary conditions. 2. What is the approximate shape of the potential U(x) in which this particle is confined? Wave function is symmetric. Wavelength is shorter in the middle. U(x) (a) U(x) (b) U(x) (c) E E E Not symmetric KE smaller in middle ∆x ∆x ∆xLecture 12, p 6 BoundBound StateState Properties:Properties: ExampleExample Let’s reinforce your intuition about the properties of bound state wave functions with this example: Through nano-engineering, one can create a step in the potential seen by an electron trapped in a 1D structure, as shown below. You’d like to estimate the wave function for an electron in the 5th energy level of this potential, without solving the SEQ. The actual wavefunction depends strongly on the parameters Uo and L. Qualitatively sketch a possible 5th wave function: U= ∞ U= ∞ Consider these features of ψ: E5 1: 5th wave function has __ zero-crossings. Uo 2: Wave function must go to zero at ____ and 0 L x ____ . ψ 3: Kinetic energy is _____ on right side of well, so the curvature of ψ is ______ there. x Lecture 12, p 7 BoundBound StateState Properties:Properties: SolutionSolution Let’s reinforce your intuition about the properties of bound state wave functions with this example: Through nano-engineering, one can create a step in the potential seen by an electron trapped in a 1D structure, as shown below. You’d like to estimate the wave function for an electron in the 5th energy level of this potential, without solving the SEQ. The actual wavefunction depends strongly on the parameters Uo and L. Qualitatively sketch a possible 5th wave function: U= ∞ U= ∞ Consider these features of ψ: E5 1: 5th wave function has __4 zero-crossings. Uo 2: Wave function must go to zero at ____x = 0 and 0 L x ____x = L . ψ 3: Kinetic energy is lower_____ on right side of well, so the curvature of ψ is ______smaller there. x The wavelength is longer. ψ and dψ/dx must be continuous here. Lecture 12, p 8 Particle in a Box U(x) As a specific important example, consider ∞ ∞ a quantum particle confined to a region, U = 0 for 0 < x < L 0 < x < L , by infinite potential walls. We call this a “one-dimensional (1D) box”. U = ∞ everywhere else 0 L This is a basic problem in “Nano-science”. It’s a simplified (1D) model of an electron confined in a quantum structure (e.g., “quantum dot”), which scientists/engineers make, e.g. , at the UIUC Microelectronics Laboratory. Quantum dots www.kfa-juelich.de/isi/ newt.phys.unsw.edu.au We already know the form of ψ when U = 0: sin(kx) or cos(kx). However, we can constrain ψ more than this Lecture 12, p 9 Particle in Infinite Square Well Potential 2π n π ψ n()sinxkx∝=() n sin x = sin x for0 ≤≤ xL ψ(x) λn L n=2 n=1 n=3 nλn = 2L 0 L x ℏ2 d2 ψ ( x ) −n +Ux()()ψ x = E ψ () x 2m dx 2 n n n The discrete E n are known as “energy eigenvalues ”: electron U = ∞ U = ∞ En p2 h 21.505 eVnm⋅ 2 n=3 E = = = n 2m 2 m λ2 λ 2 n n n=2 2 2 h n=1 EEnn =1 where E 1 ≡ 2 8mL 0 L x Lecture 11, p 10 Act 2 1. An electron is in a quantum “dot”. If we U= ∞∞∞ U= ∞∞∞ decrease the size of the dot, the ground En state energy of the electron will n=3 a) decrease n=2 b) increase n=1 c) stay the same 0 L x 2. If we decrease the size of the dot, the difference between two energy levels (e.g., between n = 7 and 2) will a) decrease b) increase c) stay the same Lecture 12, p 11 Solution 1. An electron is in a quantum “dot”. If we U= ∞ U= ∞ decrease the size of the dot, the ground En state energy of the electron will n=3 a) decrease n=2 h2 b) increase n=1 E1 = 2 c) stay the same 8mL 0 L x The uncertainty principle, once again! 2. If we decrease the size of the dot, the difference between two energy levels (e.g., between n = 7 and 2) will a) decrease b) increase c) stay the same Lecture 12, p 12 Solution 1. An electron is in a quantum “dot”. If we U= ∞ U= ∞ decrease the size of the dot, the ground En state energy of the electron will n=3 a) decrease n=2 h2 b) increase n=1 E1 = 2 c) stay the same 8mL 0 L x The uncertainty principle, once again! 2. If we decrease the size of the dot, the difference between two energy levels (e.g., between n = 7 and 2) will a) decrease 2 En = n E1 b) increase E7 -E2 = (49 – 4)E 1 = 45E 1 c) stay the same Since E 1 increases, so does ∆E. Lecture 12, p 13 “Quantum Confinement” – size of material affects “intrinsic” properties M. Nayfeh (UIUC) : Discrete uniform Si nanoparticles • Transition from bulk to molecule-like in Si • A family of magic sizes of hydrogenated Si nanoparticles • No magic sizes > 20 atoms for non- hydrogenated clusters • Small clusters glow : color depends on size 1 nm 1.67 nm 2.15 nm 2.9 nm Blue Green Yellow Red • Used to create Si nanoparticle microscopic laser: Probabilities Often what we measure in an experiment is the probability density, | ψ(x)| 2. nπ 2 2 2 nπ Probability per Wavefunction = ψ (x )= N sin x unit length ψ n (x )= N sin x Probability amplitude n L L (in 1-dimension) ψ |ψ| 2 U= ∞ U= ∞ n=1 0 L x 0 L x 2 Probability ψ |ψ| density = 0 node n=2 0 L x 0 L x ψ |ψ| 2 n=3 0 L x 0 L x Lecture 12, p 15 Probability Example Consider an electron trapped in a 1D well with L = 5 nm . Suppose the electron is in the following state: |ψ| 2 N2 2 2 2 nπ ψ n (x )= N sin x L 0 5 nm x a) What is the energy of the electron in this state (in eV)? b) What is the value of the normalization factor squared N 2? c) Estimate the probability of finding the electron within ±0.1 nm of the center of the well? (No integral required.
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