Quantum Mechanics Overview and Exam Preparations
Explore the realm of Quantum Mechanics with insights into differential equations, wave functions, and probability distributions. Get ready for the upcoming midterm exam covering waves as particles, particles as waves, and quantum mechanics. Stay engaged with Professor Pui Lam's lectures and be prepared to tackle challenging concepts in physics. Don't miss out on valuable resources and study materials to enhance your understanding of this intriguing subject.
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Midterm I, Physics 274, Fall 2017 Average = 73 Low =24 High = 100 (2) For those doing well, keep up the good work and do not slack off (midterm, final and homework remain) Please talk to me if your score is below 50%
Announcements The four powerpoint lectures of Professor Pui Lam are now posted on the course web page. I greatly appreciate his efforts and will try to figure out some way to do some of the iclicker exercises he missed (many freebies !!). Next Midterm: Monday November 13th will cover waves as particles, particles as waves and quantum mechanics. (Hope there are no major conflicts with this date).
Quantum Mechanics V The most incomprehensible thing about the world is that it is comprehensible. Albert Einstein Quiz Review of particle in a box 5
Q24.0 The name of this differential equation is: (A) Einstein s last equation (B) deBroglie s broken equation (C)Space-independent Schroedinger Equation (D) Time-independent Schroedinger Equation (E) Bound-state Schroedinger s Equation 6
Q24.0 The name of this differential equation is: (A) Einstein s last equation (B) deBroglie s broken equation (C)Space-independent Schroedinger Equation (D) Time-independent Schroedinger Equation (E) Bound-state Schroedinger s Equation Describes a free particle of mass m 7
Review: The Schrdinger equation in 1-D In a one-dimensional model, a quantum-mechanical particle is described by a wave function (x, t). [QM: remember point particles are waves] The one-dimensional Schr dinger equation for a free particle of mass m is The presence of i (the square root of 1) in the Schr dinger equation means that wave functions are always complex functions. The square of the absolute value of the wave function, | (x, t)|2, is called the probability distribution function. | (x, t)|2 dx tells us about the probability of finding the particle somewhere between location x and x+dx at time t . How do you calculate the total probability of finding the particle anywhere? Warning: | (x, t)|2 is not a probability, | (x, t)|2 dx is. 8
Q24.1 A solution to this differential equation is: (A) A cos(kx) (B) A e-kx (C)A sin (kx) (D)(B & C) (E)(A & C) 9
Q24.1 A solution to this differential equation is: (A) A cos(kx) (B) A e-kx (C)A sin (kx) (D)(B & C) (E)(A & C) Ans: E Both (A) and (C) are solutions. 10
Q24.2 = ( ) cos x A kx Condition on k just means that (p2)/2m = E. V=0, so E= KE = mv2 = p2/2m The total energy of the electron is: A. Quantized according to En = (constant) x n2, n= 1,2, 3, B. Quantized according to En = const. x (n) C. Quantized according to En = const. x (1/n2) D. Quantized according to some other condition but don t know what it is. E. Not quantized, energy can take on any value. 11
Q24.2 = ( ) cos x A kx 2 2 ( 2 ) x = ( ) E x 2k2 2m 2 m makes sense, because x = E p = k Condition on k is just saying that (p2)/2m = E. V=0, so E= KE = mv2 = p2/2m The total energy of the electron is: A. Quantized according to En = (constant) x n2, n= 1,2, 3, B. Quantized according to En = const. x (n) C. Quantized according to En = const. x (1/n2) D. Quantized according to some other condition but don t know what it is. E. Not quantized, energy can take on any value. Ans: E - No boundary, energy can take on any value. 12
Q25.0 Compare a free particle and a particle confined inside a box ( particle in a box ). According to quantum theory, which of the following statements is true? A. The possible energies of the free particle are quantized. B. The possible energies of the free particle and the particle in the box are quantized. C. Only the possible energies of the particle in the box are quantized. 13
Q25.0 Compare a free particle and a particle confined inside a box ( particle in a box ). According to quantum theory, which of the following statements is true? A. The possible energies of the free particle are quantized. B. The possible energies of the free particle and the particle in the box are quantized. C. Only the possible energies of the particle in the box are quantized. 14
Review: Infinite Potential Well (particle in a box) E quantized by B. C. s 2 L = n What is E? A. can be any value (not quantized). B. C. E. D. Does this L dependence make sense? 15
Q25.1 Given that the ground state (n=1) energy of a particle in a box is 10 eV. What is the energy of the first excited level (n=2)? A. 20 eV B. 30 eV C. 40 eV D. 50 eV 16
Q25.1 Given that the ground state (n=1) energy of a particle in a box is 10 eV. What is the energy of the first excited level (n=2)? A. 20 eV B. 30 eV C. 40 eV D. 50 eV Note n2 dependence 17
Q25.2 Given that length of the box is 10-10m. What the ground state (n=1) wavelength of a particle in this box? A.10-10m B.0.5x10-10m C.2x10-10m D.4x10-10m 18
Q25.2 Given that length of the box is 10-10m. What the ground state (n=1) wavelength of a particle in this box? Remember = 2 L = C A.10-10m B.0.5x10-10m C.2x10-10m D.4x10-10m What are the wavefunctions for the particle in the box ? 19
Q25.3 Given that the ground state (n=1) wavelength of a particle in this box is 10-10m. What is the wavelength of the first excited state (n=2)? A.10-10m B.0.5x10-10m C.2x10-10m D.4x10-10m 20
Q25.3 Given that the ground state (n=1) wavelength of a particle in this box is 10-10m. What is the wavelength of the first excited state (n=2)? A.10-10m B.0.5x10-10m C.2x10-10m D.4x10-10m 21
Review: Infinite Potential Well (particle in a box) Probability of finding particle at a specific x-location? Probability of finding particle at a specific x-location? 22
Infinite Potential Well (particle in a box) Probability of finding particle at a specific x-location? 24
