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**Unformatted text preview: **Lecture Notes on Quantum Mechanics - Part II Yunbo Zhang Instituter of Theoretical Physics, Shanxi University Abstract In this chapter we solve the stationary Schr odinger equation for various types of one dimensional potentials, including square potential well, harmonic oscillator, scattering of potential barrier, and -potential. 1 Contents I. Square Potential Well 4 A. An idealized case - Infinite square potential well 4 1. Solution 4 2. Energy quantization 5 3. Wave function normalization 6 4. Shift of origin of coordinate 7 B. Square potential well of finite depth 8 1. Solution 8 2. Even parity case 9 3. Odd parity case 9 4. Graphical solution of the energy defining equation 10 II. Generic Problem of Quantum Mechanics 12 A. Properties of wave functions 13 B. Construction of the most general solution 14 C. An example 15 III. The Harmonic Oscillator 17 A. Algebraic method: ladder operator 18 1. Commutator 18 2. Ladder operators 19 3. Normalization algebraically: find A n 23 B. Analytic method: differential equation 25 1. Discussion on the energy quantization 28 2. Discussion on the wave function 28 IV. The Free Particle: Revisited 32 V. Penetration of Potential Barrier 38 A. Case I: Barrier penetration E < V 39 1. Boundary conditions 41 B. Case II: Barrier penetration E > V 43 2 C. Case III: Bound states in finite potential well- V < E < 43 D. Case IV: Scattering from square well- V < < E 44 VI. The -function potential 45 A. Bound states and scattering states 45 B. From square potential to -potential 47 C. Bound states in potential well ( E < 0) 48 D. Scattering states in -potential well ( E > 0) 50 E. -function barrier 52 VII. Summary on Part II 53 3 I. SQUARE POTENTIAL WELL We consider a particle moving in a square potential well (Figure 1) V ( x ) = , < x < a V , x < ,x > a and in this case the stationary Schr odinger equation- ~ 2 2 m d 2 dx 2 ( x ) + V ( x ) ( x ) = E ( x ) can be solved exactly. The physics here is the one-dimensional material-wave propagation in a potential well. We will confine our discussions to cases with energy E less than V . A. An idealized case - Infinite square potential well First we solve the idealized model of above potential, that is when V + , a infinite square potential well 1. Solution For the potential V ( x ) = , < x < a + , x < ,x > a it is easy to know- ~ 2 2 m d 2 dx 2 ( x ) = E ( x ) for 0 < x < a ( x ) = 0 for x < ,x > a Let k 2 = 2 mE/ ~ 2 , we have d 2 dx 2 ( x ) + k 2 ( x ) = 0 ( x ) = A sin( kx + ) A and are two constants to be determined shortly. 4 a x V(x) V FIG. 1: One dimensional square potential well. 2. Energy quantization According to the natural boundary conditions, ( x ) should be continuous at x = 0 and x = a . So (0) = ( a ) = 0 which tells us = 0, and ka = n, with n = 1 , 2 , 3 (because A = 0 and n = 0 give trivial, non-normalizable solution ( x ) = 0 . Furthermore n =- 1 ,...

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