1. Other examples of one-dimensional motion
PHY 741 Quantum Mechanics
12-12:50 PM MWF Olin 103
Plan for Lecture 9:
Continue reading Chapter #7 in Shankar; Eigenstates of the one-dimensional Schrödinger equation
Harmonic oscillator
Comment on HW #5
Other examples of one-dimensional motion
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PHY 741 Fall Lecture 9

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4. One-dimensional harmonic oscillator
Define:
Note that:
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PHY 741 Fall Lecture 9

5. Representation of the position and momentum operators in terms of the energy eigenstates of the harmonic oscillator:
n= …..
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PHY 741 Fall Lecture 9

6. Representation of the raising and lowering operators in terms of the energy eigenstates of the harmonic oscillator:
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PHY 741 Fall Lecture 9

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PHY 741 Fall Lecture 9

8. Example from statistical mechanics:
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PHY 741 Fall Lecture 9

9. Example: Particle of mass m confined within an finite square well:
Comment on HW #5
Example: Particle of mass m confined within an finite square well:
V(x) V0 V0 x a -a
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PHY 741 Fall Lecture 9

10. After some algebra, we found that the energy eigenvalues E were solutions of transcendental equations:
For even parity
For odd
parity
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PHY 741 Fall Lecture 9

11. Solution using maple:
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PHY 741 Fall Lecture 9

12. Another example of a one-dimensional system
Consider an electron moving in a one-dimensional model potential (Kronig and Penney, Proc. Roy. Soc. (London) 130, 499 (1931) V0 a
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PHY 741 Fall Lecture 9

13. V0 a
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PHY 741 Fall Lecture 9

14. V0 a
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PHY 741 Fall Lecture 9

15. V0 a
Band gap
E(k)
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PHY 741 Fall Lecture 9

16. Consider a single potential well Spectrum
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PHY 741 Fall Lecture 9

17. Consider a periodic potential well system
Spectrum
Consider a periodic potential well system E V0 a
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PHY 741 Fall Lecture 9

18. V0 a b
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PHY 741 Fall Lecture 9

19. V0 a b
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20. V0 a b
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21. Forbidden states f Forbidden states v 9/15/2017
PHY 741 Fall Lecture 9

22. f
Band gap
Band gap
ka/p
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PHY 741 Fall Lecture 9