Physical chemistry (quantum) hw

2023/01/19Chem 113A Problem Set 02: Lectures 04–05
Due 2023/01/26 @ 10 PM
We encourage you to work with others in the class on the problem set, but the end product must represent your
effort. Useful integrals can be found on the last page.
Part 1 (To be done in Discuss Section. If you do not finish it in section, please finish at home)
1) Particle in a finite box
For a particle in a finite box, where the potential energy outside the box do not go to infinity, the
wavefunction does not go to zero at the edge (as shown in the picture above). For particle in a finite box,
the potential energy is described as
𝐿
𝐿
0, − ≤ 𝑥 ≤
𝑉(𝑥) = {
2
2
𝑈0 , otherwise
𝐿
The center of the box is at 𝑥 = 0 and the edge of the box is at 𝑥 = ± 2. In this situation, the wave function
for the ground state (n=1) is
𝜓(𝑥) = 𝐴𝑒 −𝑎|𝑥|
where
2𝑚(𝑈0 − 𝐸)
𝑎=√
ℏ2
𝑈0 is the potential energy outside the box, 𝐸 is the energy of the particle, and 𝐸 < 𝑈0 a. Set up the integral that calculates the probability of finding the particle inside the box. pg. 1 Problem Set 02 Due 01/26 b. The probability of finding the particle inside the box plus the probability of finding the particle outside the box equals one, because the particle must be either inside or outside the box. Express the probability of finding the particle outside the box in terms of your answer from part a. c. To deal with the absolute value, we’ll use the following relation 𝑎 𝑎 ∫𝑒 −𝑐|𝑥| −𝑎 𝑑𝑥 = 2 ∫ 𝑒 −𝑐𝑥 𝑑𝑥 0 Calculate the probability of finding the particle outside the box in terms of 𝐴, 𝑎, and 𝐿 𝐴2 Note: Using the normalization condition, 𝑎 = 1 (You can try to prove to yourself that this is true.) d. For classical physics, if the potential energy at the edge of the box (U0) is greater than the energy of the particle, is it possible for the particle to be outside the box? What about quantum mechanics? pg. 2 Problem Set 02 Due 01/26 e. What happens to the probability of finding the particle outside the box as 𝑈0 goes to infinity? 2) Operators Evaluate 𝑔 = 𝐴̂𝑓, where 𝐴̂ and 𝑓 are given below: 3 𝑑 a. 𝐴̂ = 𝑑𝑥 3 ; 𝑓 = 𝑒 −𝑎𝑥 1 b. 𝐴̂ = ∫0 𝑑𝑥 ; 𝑓 = 𝑥 3 − 2𝑥 + 3 𝜕 𝜕 𝜕 c. 𝐴̂ = 𝜕𝑥 + 𝜕𝑦 + 𝜕𝑧 ; 𝑓 = 𝑥 3 𝑦 2 𝑧 4 pg. 3 Problem Set 02 Due 01/26 3) Superposition of Particle in a 1-D Box For a particle in a box, supposed our particle is described as the sum of the 𝑛 = 1 wavefunction and the 𝑛 = 2 wavefunction: 𝜓(𝑥) = 𝑐1 𝜓𝑛=1 (𝑥) + 𝑐2 𝜓𝑛=2 (𝑥) where 2 𝑛𝜋 𝜓𝑛 = √ sin ( 𝑥) 𝐿 𝐿 a. Furthermore, let’s say 𝑐1 = 𝑐2 . 2 𝜋 2 2𝜋 𝜓(𝑥) = 𝑐1 √ sin ( 𝑥) + 𝑐1 √ sin ( 𝑥) 𝐿 𝐿 𝐿 𝐿 Normalize the wavefunction 𝜓(𝑥) Useful Integrals 𝐿 𝜋 2𝜋 ∫ sin ( 𝑥) sin ( 𝑥) 𝑑𝑥 = 0 𝐿 𝐿 0 𝐿 𝑛𝜋 𝐿 ∫ sin2 ( 𝑥) 𝑑𝑥 = 𝐿 2 0 pg. 4 Problem Set 02 Due 01/26 Part 2 (Take-home) 1) Particle in a 2-D Box In recent years, a lot of research have been focused on 2-D materials such as perovskites, MoS2, or graphene sheet. These materials are typically only a few atoms thick, so they can be modeled as particle in a 2-D box. The Hamiltonian for a 2-D box is ̂=− 𝐻 ℏ2 𝜕 2 𝜕2 + ( ) 2𝑚 𝜕𝑥 2 𝜕𝑦 2 a. Show that 𝜓(𝑥, 𝑦) = 𝑛𝑦 𝜋 2 𝑛𝑥 𝜋 sin ( 𝑥) sin ( 𝑦) 𝐿𝑥 𝐿𝑦 𝐿𝑥 𝐿𝑦 is an eigenfunction of the 2-D Hamiltonian, where 𝐿𝑥 and 𝐿𝑦 is the length of the box in the x- and ydirections, respectively, and 𝑛𝑥 and 𝑛𝑦 are the quantum numbers. What is the energy for the particle in a 2-D box as a function of 𝐿𝑥 , 𝐿𝑦 , 𝑛𝑥 and 𝑛𝑦 . pg. 5 Problem Set 02 Due 01/26 b. If our box is a square (𝐿𝑥 = 𝐿𝑦 ), plot the energy level diagram for the first 8 energy levels for a particle in a 2-D box. c. Use Figure 1 and Figure 2 on this website (half way down the webpage): https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplement al_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Mechanics/05.5%3A_Particle_in_Boxes/Parti cle_in_a_2-Dimensional_Box to visualize the wave function for the first 8 energy levels. You can drag the surface plots to rotate and look at the wave functions from different angles. From the energy level diagram, you see that (𝑛𝑥 = 2, 𝑛𝑦 = 1) and (𝑛𝑥 = 1, 𝑛𝑦 = 2) are degenerate (i.e., they have the same energy). From the plots of the wavefunctions, does it make sense that these two energy levels are degenerate? pg. 6 Problem Set 02 Due 01/26 d. How can one break degeneracies for a particle in a 2-D box? (i.e., how can one make (𝑛𝑥 = 2, 𝑛𝑦 = 1) and (𝑛𝑥 = 1, 𝑛𝑦 = 2) have different energies? e. For a particle in a 2D-box, is it possible to measure its x-position and y-momentum simultaneously? In other words, what is the uncertainty relationship Δ𝑥Δ𝑝𝑦 equal to? 𝜕 𝑥̂ = 𝑥; 𝑝̂𝑦 = −𝑖ℏ 𝜕𝑦 pg. 7 Problem Set 02 Due 01/26 2) Angular Momentum Operator The angular momentum operators are defined as followed: 𝜕 𝜕 𝜕 𝜕 𝜕 𝜕 𝐿̂𝑥 = −𝑖ℏ (𝑦 − 𝑧 ) ; 𝐿̂𝑦 = −𝑖ℏ (𝑧 − 𝑥 ) ; 𝐿̂𝑧 = −𝑖ℏ (𝑥 −𝑦 ) 𝜕𝑧 𝜕𝑦 𝜕𝑥 𝜕𝑧 𝜕𝑦 𝜕𝑥 ̂ ̂ ̂ ̂ ̂ ̂ a. Calculate the following commutators, [𝐿𝑥 , 𝐿𝑦 ], [𝐿𝑦 , 𝐿𝑧 ], and [𝐿𝑧 , 𝐿𝑥 ]. b. What do the commutators say about the ability to measure the components of angular momentum simultaneously? pg. 8 2023/01/18 Chem 113A Lecture 04: Wavefunctions and Operators Objectives: • Know what a wavefunction and an operator are. o Suggested readings: 3.2, 3.3, 3.4, 3.6, 4.1, 4.2 o Suggested textbook problems: 3-1,3-2, 3-5, 3-6, 4-1, 4-2, 4-3, 4-5 Time-Independent Schrödinger Equation ̂ Ψ = 𝐸Ψ 𝐻 Postulate 1 The state of quantum-mechanical system is completely specified by a function Ψ(𝑟, 𝑡) that depends on the coordinates of the particle and on time. This function, called the wavefunction of state function, has the important property that Ψ ∗ (𝑟, 𝑡)Ψ(𝑟, 𝑡)𝑑𝑥𝑑𝑦𝑑𝑧 is the probability that the particle lies in the volume element 𝑑𝑥𝑑𝑦𝑑𝑧 located at 𝑟 and at time 𝑡. Particle-in-a-box: Page 1 Lecture 04 2023/01/18 Example What is the probability of finding the particle between 𝑥 = 0 and 𝑥 = 𝐿/4 for the ground state of a particle-in-a-box? Useful integral 𝐿/4 𝑛𝜋𝑥 𝐿 𝐿 𝑛𝜋 ) 𝑑𝑥 = − sin ( ) 𝐿 8 4𝑛𝜋 2 ∫ sin2 ( 0 Two requirements for Postulate 1: 1. The wavefunction must be well behaved. 2. The wavefunction must be normalized. Page 2 Lecture 04 2023/01/18 Example Let Ψ(𝑥) = 𝑖 − 𝑥, where −1 ≤ 𝑥 ≤ 1. Is Ψ(𝑥) normalized? If not, normalized Ψ(𝑥) Postulate 2 To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics. Operator: a symbol that tells you to do something to whatever follows the symbol. Example Let 𝑓(𝑥, 𝑡) = cos(𝑡) sin(𝑥). Evaluate 𝐴̂𝑓(𝑥, 𝑡), where 𝐴̂ is given below: 𝜕 a. 𝐴̂ = 𝜕𝑥 𝜕2 𝜕 c. 𝐴̂ = 𝜕𝑥 2 + 𝜕𝑡 + 3 b. 𝐴̂ = 𝑥 Page 3 Lecture 04 2023/01/18 Linear Operator An operator is linear if… Example Determine whether the following operators are linear or nonlinear: a. 𝐴̂ = 𝑑/𝑑𝑥 b. 𝐴̂ = SQRT Order of operation with operators 𝑑 Example Let 𝐴̂ = 𝑥, 𝐵̂ = 𝑑𝑥, and 𝑓(𝑥) = sin(𝑥). Perform the following operations: a. 𝐴̂𝐵̂ 𝑓(𝑥) b. 𝐵̂ 𝐴̂𝑓(𝑥) c. 𝐵̂ 2 𝑓(𝑥) Page 4 2023/01/20 Chem 113A Lecture 5: Commutator, Eigenfunctions, and Eigenvalues Objective: ● Know how to calculate the commutator of two operators and what it means. ● Know how to identify whether a function is an eigenfunction of an operator. ● Know how to calculate eigenvalues given an eigenfunction and an operator. o Suggested reading: 3.3 – 3.6, 4.3, 4.4 o Suggested textbook problems: 4-11 – 4-17 The Commutator of Two Operators Example Evaluate the commutator [𝑝̂𝑥 , 𝑥] If two operators do not commute, then their corresponding observable quantities do not have simultaneously well-defined values. 2 1 𝜎𝐴2 𝜎𝐵2 ≥ − ( 4 ∫ 𝜓 ∗ [𝐴̂, 𝐵̂ ]𝜓 𝑑𝑥) 𝑎𝑙𝑙 𝑠𝑝𝑎𝑐𝑒 Page 1 Lecture 05 2023/01/20 Example Use the commutator of the position operator and the momentum operator to derive the Heisenberg uncertainty principle. Postulate 3 In any measurement of the observable associated with the operator, 𝐴̂, the only values that will ever be observed are the eigenvalues 𝑎, which satisfy the eigenvalue equation 𝐴̂𝜓𝑎 = 𝑎𝜓𝑎 𝑑𝑛 Example Is 𝑒 𝑎𝑥 an eigenfunction of 𝑑𝑥 𝑛. If so, what is the eigenvalue? Time-Independent Schrödinger Equation is an example of an eigenvalue/eigenfunction problem. Page 2 Lecture 05 2023/01/20 Particle-in-a-Box Potential Energy Boundary Conditions Solving the Schrödinger Equation Three Cases: 𝐾 = 0, 𝐾 > 0, and 𝐾 < 0 Case 1: 𝐾 = 0 Page 3 Lecture 05 2023/01/20 Particle-in-a-Box Solution Page 4 2023/01/23 Chem 113A Lecture 06: Postulate 3, Hermitian Operators, Expanding Wavefunctions Objective: ● Know the following terms: Hermitian operators, orthonormal, Fourier coefficients. ● Know how to expand a wavefunction in terms of a set of eigenfunctions. ● Know how to calculate the variance of an operator o Suggested readings: 4.3 – 4.8 o Suggested textbook problems: 4-22 – 4-24 Postulate 3 In any measurement of the observable associated with the operator, 𝐴̂, the only values that will ever be observed are the eigenvalues, 𝑎, which satisfy the eigenvalue equation 𝐴̂𝜓𝑎 = 𝑎𝜓𝑎 Example: Particle-in-a-box Hermitian Operators Definition: Why Hermitian operators are special? Page 1 Lecture 06 2023/01/23 Example: Particle-in-a-box An analogy using the Cartesian Coordinate ** The probability of obtaining a certain value of an observable in a measurement is given by the Fourier coefficient ** Page 2 Lecture 06 2023/01/23 Example Expand 1 30 2 Ψ(𝑥) = ( 5 ) 𝑥(𝐿 − 𝑥), 0 ≤ 𝑥 ≤ 𝐿 𝐿 in terms of the orthonormal complete set of eigenfunctions of a particle in a box. 𝐿 𝑛𝜋𝑥 𝐿 3 ∫ 𝑥(𝐿 − 𝑥) sin ( ) 𝑑𝑥 = 2 ( ) [1 − cos(𝑛𝜋)] 𝐿 𝑛𝜋 0 If we measure the total energy of Ψ(𝑥), what is the probability of obtaining 𝐸 = 𝐸𝑛=1 ? Page 3 Lecture 06 2023/01/23 Example If we measure the total energy of 𝜓𝑛=1, what is the probability of obtaining 𝐸 = 𝐸𝑛=2 ? Postulate 3 and Uncertainty Principle If two operators commute, they have the same set of eigenfunctions… Page 4