Clicker Questions Lecture Slides Professor John Price, Spring 2019 Physics 2130 Foundations of Modern Physics Quantum Mechanics Part II: The Schrödinger Equation in One Dimension Clicker Questions Lecture Slides Professor John Price, Spring 2019

The Schrödinger Equation in One Dimension TZD Chapter 7 Time-dependent Schrödinger equation Time-independent Schrödinger equation and stationary states Particle in a box (square-well potential) Finite square-well potential Harmonic oscillator Tunneling Superposition Clicker a. start / stop b, hide/ unhide c. ppt-1page fwd d. ppt 1 pg back e. select correct answer

What system corresponds to each sketch? continuum E=0 A B C D E F What system corresponds to each sketch? Some options: harmonic oscillator, free particle, hydrogen, spin-1/2 particle in a B field, infinite square well, finite square well Every system must have a ground state. . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c

Suppose 𝜕 2 𝑓(𝑥) 𝜕 𝑥 2 =− 𝑏 2 𝑓(𝑥) where b2 is a positive real number Suppose 𝜕 2 𝑓(𝑥) 𝜕 𝑥 2 =− 𝑏 2 𝑓(𝑥) where b2 is a positive real number. Which functions below are solutions to this equation? sin⁡(𝑏𝑥) cos⁡(𝑏𝑥) 𝑒 𝑖𝑏𝑥 𝑒 −𝑖𝑏𝑥 3 cos 𝑏𝑥 +𝑖 𝑒 3 𝑒 𝑖𝑏𝑥 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 46 Physics 2130, Fall 2019

Suppose 𝜕 2 𝑓(𝑥) 𝜕 𝑥 2 = 𝑎 2 𝑓(𝑥) where a2 is a positive real number Suppose 𝜕 2 𝑓(𝑥) 𝜕 𝑥 2 = 𝑎 2 𝑓(𝑥) where a2 is a positive real number. Which functions below are solutions to this equation? 𝑠𝑖𝑛⁡(𝑎𝑥) 𝑐𝑜𝑠⁡(𝑎𝑥) 𝑒 𝑖𝑎𝑥 𝑒 −𝑖𝑎𝑥 𝑒 𝑎𝑥 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 47 Physics 2130, Spring 2019

Suppose 𝜕 2 𝑓(𝑥) 𝜕 𝑥 2 = −𝑘 2 𝑓(𝑥) where k2 is a positive real number. Some solutions are: 𝑒 𝑖𝑘𝑥 , 𝑒 −𝑖𝑘𝑥 , 𝑠𝑖𝑛⁡(𝑎𝑥), 𝑐𝑜𝑠⁡(𝑎𝑥) But I thought there were only suppose to be 2 solutions! What’s going on? They lied to me. There are supposed to be at least 2 solutions. There are only 2 independent solutions. Changing i to −i does not count, likewise sin to cos. . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 48 Physics 2130, Spring 2019

it depends, this is a bad question running wave moving towards +x 𝜓 𝑥 =𝐴𝑒 𝑖𝑘𝑥 ψ is what kind of wave? standing wave it depends, this is a bad question running wave moving towards +x running wave moving towards -x . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 49 Physics 2130, Spring 2019

it depends, this is a bad question running wave moving towards +x 𝜓 𝑥 =𝐴𝑐𝑜𝑠(𝑘𝑥) ψ is what kind of wave? standing wave it depends, this is a bad question running wave moving towards +x running wave moving towards -x . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 50 Physics 2130, Spring 2019

it depends, this is a bad question running wave moving towards +x 𝜓 𝑥 =𝐴𝑒 𝑖𝑘𝑥 +𝐵 𝑒 −𝑖𝑘𝑥 ψ is what kind of wave? standing wave it depends, this is a bad question running wave moving towards +x running wave moving towards -x . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 51 Physics 2130, Spring 2019

Is n=0 a zero-energy solution? Infinite square well 𝐸 𝑛 = 𝑛 2 ℏ 2 𝜋 2 2𝑚 𝑎 2 Is n=0 a zero-energy solution? yes no . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 52 Physics 2130, Spring 2019

Consider the modes of a guitar string. Is it true that 𝑓 𝑛 = 𝑛 2 ∗𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 ? A) Yes. Why? What is the same about the guitar string and the infinite square well? B) No. Why not? What is different? . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 53 Physics 2130, Spring 2019

Which is one is a particle in a box? continuum E=0 A B C D E F . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Which is one is a particle in a box? Clicker Slide 54 Physics 2130, Spring 2019

Sketch 𝜓 𝑥 and 𝜓 ∗ 𝜓 for the case n=3. What is the probability Infinite square well Sketch 𝜓 𝑥 and 𝜓 ∗ 𝜓 for the case n=3. What is the probability to find the particle between 0 and a/3? 1 1/4 1/2 1/3 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 55 Physics 2130, Spring 2019

Sketch 𝜓 ∗ 𝜓 for n=1. What is 𝑥 ? Infinite square well Sketch 𝜓 ∗ 𝜓 for n=1. What is 𝑥 ? a a/4 a/2 a/3 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 56 Physics 2130, Spring 2019

Sketch 𝜓 ∗ 𝜓 for n=3. What is 𝑥 ? Infinite square well Sketch 𝜓 ∗ 𝜓 for n=3. What is 𝑥 ? a a/4 a/2 a/3 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 57 Physics 2130, Spring 2019

𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥 U x Region I Region II Potential step 𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥 Suppose 𝐸>U. How is E related to k? 𝐸−𝑈= ℏ 2 𝑞 2 2𝑚 𝐸−𝑈= ℏ 2 𝑘 2 2𝑚 𝐸= ℏ 2 𝑞 2 2𝑚 𝐸= ℏ 2 𝑘 2 2𝑚 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 58 Physics 2130, Spring 2019

𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥 U x Region I Region II Potential step 𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥 Suppose 𝐸>U. How is E related to q? 𝐸−𝑈= ℏ 2 𝑞 2 2𝑚 𝐸−𝑈= ℏ 2 𝑘 2 2𝑚 𝐸= ℏ 2 𝑞 2 2𝑚 𝐸= ℏ 2 𝑘 2 2𝑚 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 59 Physics 2130, Spring 2019

𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥 U x Region I Region II Potential step 𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥 Suppose 𝐸>U. Which expression is correct? 𝐴=𝐵 𝑖𝑘𝐴=𝑖𝑞𝐵 1+𝐴=𝐵 −𝑖𝑘+𝐴=𝑖𝑞 𝐴+𝐵=1 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 60 Physics 2130, Spring 2019

𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥 U x Region I Region II Potential step 𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥 Suppose 𝐸>U. Which expression is correct? 𝐴=2𝐵 𝑘(1−𝐴)=𝑞𝐵 𝐴=𝐵 −𝑖𝑘+𝐴=𝑖𝑞 𝐴+𝐵=1 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 61 Physics 2130, Spring 2019

Potential step 𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝛼𝑥 U x Region I Region II 𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 Suppose 0<E<U. What is the form of the wavefunction in region II? (α and q are positive real). 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝛼𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝛼𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 −𝛼𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 −𝑖𝑞𝑥 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 62 Physics 2130, Spring 2019

𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥 U x Region I Region II Potential step 𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥 Suppose 𝐸≫U. Which expression is correct? 𝐴=𝐵 𝐴=1, 𝐵=0 𝐴=0, 𝐵=1 𝐴=0, 𝐵=−1 𝐴=−𝐵 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 63 Physics 2130, Spring 2019

Finite square well In the ground state the particle has a small probability to be found outside the potential well. 𝑈 0 / 𝐸 1 (∞)=62.5 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Physics 2130, Spring 2019

Finite square well The 4th state (n=4). What is the form of 𝜓? For this problem, A and B are positive and real. Region I Region III A 𝑒 𝑎𝑥 B 𝑒 −𝑎𝑥 𝐴𝑒 −𝑎𝑥 B 𝑒 𝑎𝑥 𝐴𝑒 𝑎𝑥 -B 𝑒 −𝑎𝑥 𝐴𝑒 −𝑎𝑥 B 𝑒 −𝑎𝑥 𝐴𝑒 𝑎𝑥 -B 𝑒 𝑎𝑥 𝑈 0 / 𝐸 1 (∞)=62.5 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Region I Region III Clicker Slide 64 Physics 2130, Spring 2019

Finite square well The 4th state (n=4). What is the form of 𝜓? For this problem, A and B are positive and real. Region I Region III A 𝑒 𝑎𝑥 B 𝑒 −𝑎𝑥 𝐴𝑒 −𝑎𝑥 B 𝑒 𝑎𝑥 𝐴𝑒 𝑎𝑥 -B 𝑒 −𝑎𝑥 𝐴𝑒 −𝑎𝑥 B 𝑒 −𝑎𝑥 𝐴𝑒 𝑎𝑥 -B 𝑒 𝑎𝑥 𝑈 0 / 𝐸 1 (∞)=62.5 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 64 Physics 2130, Spring 2019

What is the form of 𝜓? x=0 is at the center of the well. Finite square well The 5th state (n=5). What is the form of 𝜓? x=0 is at the center of the well. Region II A 𝑒 𝑎𝑥 𝐴𝑒 −𝑎𝑥 𝐴𝑠𝑖𝑛(𝑘𝑥) 𝐴𝑐𝑜𝑠(𝑘𝑥) 𝑈 0 / 𝐸 1 (∞)=62.5 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Region II Clicker Slide 65 Physics 2130, Spring 2019

What is the form of 𝜓? x=0 is at the center of the well. Finite square well The 5th state (n=5). What is the form of 𝜓? x=0 is at the center of the well. Region II A 𝑒 𝑎𝑥 𝐴𝑒 −𝑎𝑥 𝐴𝑠𝑖𝑛(𝑘𝑥) 𝐴𝑐𝑜𝑠(𝑘𝑥) 𝑈 0 / 𝐸 1 (∞)=62.5 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 65 Physics 2130, Spring 2019

Finite square well The 5th state (n=5). What is the form of 𝜓? For this problem, A and B are positive and real. Region I Region III A 𝑒 𝑎𝑥 B 𝑒 −𝑎𝑥 𝐴𝑒 −𝑎𝑥 B 𝑒 𝑎𝑥 𝐴𝑒 𝑎𝑥 -B 𝑒 −𝑎𝑥 𝐴𝑒 −𝑎𝑥 B 𝑒 −𝑎𝑥 𝐴𝑒 𝑎𝑥 -B 𝑒 𝑎𝑥 𝑈 0 / 𝐸 1 (∞)=62.5 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Region I Region III Clicker Slide 66 Physics 2130, Spring 2019

Finite square well The 5th state (n=5). What is the form of 𝜓? For this problem, A and B are positive and real. Region I Region III A 𝑒 𝑎𝑥 B 𝑒 −𝑎𝑥 𝐴𝑒 −𝑎𝑥 B 𝑒 𝑎𝑥 𝐴𝑒 𝑎𝑥 -B 𝑒 −𝑎𝑥 𝐴𝑒 −𝑎𝑥 B 𝑒 −𝑎𝑥 𝐴𝑒 𝑎𝑥 -B 𝑒 𝑎𝑥 𝑈 0 / 𝐸 1 (∞)=62.5 Region I Region III Clicker Slide 66 Physics 2130, Spring 2019

Which sketch represents the spectrum of a finite square well? 𝐴 𝐵 𝐶 𝐷 A B C D E F Which sketch represents the spectrum of a finite square well? 𝐴 𝐵 𝐶 𝐷 𝐸 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 67 Physics 2130, Spring 2019

𝜓 𝑦 = 𝑒 − 𝑦 2 /2 What is 𝜕 2 𝜕 𝑦 2 𝜓 𝑦 ? y𝜓(𝑦) 𝑦 2 𝜓 𝑦 ( 𝑦 2 −1)𝜓 𝑦 ( 𝑦 2 +1)𝜓 𝑦 ) 1 2 𝑦 2 𝜓 𝑦 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 68 Physics 2130, Spring 2019

2 𝑎 † =𝑦− 𝜕 𝜕𝑦 What does 2 𝑎 † 𝜓 𝑦 mean? y𝜓(𝑦) 𝜕 𝜕𝑦 𝜓(𝑦) − 𝜕 2 𝜕𝑦 𝜓 𝑦 − 𝜕 2 𝜕𝑦 𝜓 𝑦 y𝜓 𝑦 − 𝜕 𝜕𝑦 𝜓(𝑦) y𝜓 𝑦 + 𝜕 𝜕𝑦 𝜓(𝑦) . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 69 Physics 2130, Spring 2019

2 𝑎 † 𝜓 0 𝑦 = 2y 𝜓 0 (𝑦) 2(1−y) 𝜓 0 𝑦 2𝑦 2 𝜓 0 (𝑦) 2(1− 𝑦 2 ) 𝜓 0 𝑦 2 𝑎 † 𝜓 0 𝑦 = 2y 𝜓 0 (𝑦) 2(1−y) 𝜓 0 𝑦 2𝑦 2 𝜓 0 (𝑦) 2(1− 𝑦 2 ) 𝜓 0 𝑦 . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 70 Physics 2130, Spring 2019

Infinite square well again 𝑘 𝑛 =𝑛 𝜋 𝑎 𝜓 1 𝑥 = 𝐴 1 sin 𝑘 1 𝑥 𝐸 𝑛 = 𝑛 2 ℏ 2 𝜋 2 2𝑚 𝑎 2 0 a Consider the superposition state 𝜓 1 𝑥 + 𝜓 2 𝑥 . Is this a solution to the TISE? Yes, it’s a linear and homogenous equation No . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 71 Physics 2130, Spring 2019

Infinite square well again 𝑘 𝑛 =𝑛 𝜋 𝑎 𝜓 1 𝑥 = 𝐴 1 sin 𝑘 1 𝑥 𝐸 𝑛 =ℏ 𝜔 𝑛 = 𝑛 2 ℏ 2 𝜋 2 2𝑚 𝑎 2 0 a Consider the superposition state Ψ 𝑥,𝑡 = 𝜓 1 𝑥 𝑒 −𝑖 𝜔 1 𝑡 + 𝜓 2 𝑥 𝑒 −𝑖 𝜔 2 𝑡 . Is this a solution to the TDSE? Yes, it’s a linear and homogenous equation No . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 72 Physics 2130, Spring 2019

Infinite square well again 𝑘 𝑛 =𝑛 𝜋 𝑎 𝜓 𝑛 𝑥 = 𝐴 𝑛 sin 𝑘 𝑛 𝑥 𝐸 𝑛 =ℏ 𝜔 𝑛 = 𝑛 2 ℏ 2 𝜋 2 2𝑚 𝑎 2 0 a Consider the superposition state Ψ 𝑥,𝑡 = 𝑛=1 ∞ 𝐴 𝑛 𝜓 𝑛 𝑥 𝑒 −𝑖 𝜔 𝑛 𝑡 . Is this a solution to the TDSE? Yes, it’s a linear and homogenous equation No . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 73 Physics 2130, Spring 2019

Suppose we have a set of solutions to the TISE for any U(x): TISE  TDSE Suppose we have a set of solutions to the TISE for any U(x): 𝜓 𝑛 𝑥 𝐸 𝑛 =ℏ 𝜔 𝑛 Consider the superposition state Ψ 𝑥,𝑡 = 𝑛=1 ∞ 𝐴 𝑛 𝜓 𝑛 𝑥 𝑒 −𝑖 𝜔 𝑛 𝑡 . Is this a solution to the TDSE? Yes, it’s a linear and homogenous equation No . Discuss the form of the v transformation formula . Why plus not minus . Discuss that is always smaller than c Clicker Slide 74 Physics 2130, Spring 2019