1. 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

2. 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
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3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥U 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

17. 𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥U 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

18. 𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥U 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

19. 𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥U 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

20. Potential step 𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝛼𝑥U 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

21. 𝜓 𝐼 𝑥 = 𝑒 𝑖𝑘𝑥 +𝐴 𝑒 −𝑖𝑘𝑥 𝜓 𝐼𝐼 𝑥 = 𝐵𝑒 𝑖𝑞𝑥U 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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 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

34. 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 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

35. 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 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

36. 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