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2. Show that the first two wavefunctions of the harmonic oscillator (McQuarrie Table 5.3, p. 170) are normalized and that they are orthogonal to each other. a. Vo is normalized: Y(x) = (=)" -ax²12 b. ₁ is normalized: Y₁(x) = (40(²) 14 -xx²/2 xe c. o and ₁ are orthogonal: of 10000

2. Show that the first two wavefunctions of the harmonic oscillator (McQuarrie Table 5.3, p. 170) are normalized and that they are orthogonal to each other. a. Vo is normalized: Y(x) = (=)" -ax²12 b. ₁ is normalized: Y₁(x) = (40(²) 14 -xx²/2 xe c. o and ₁ are orthogonal: of 10000

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Subject is physical chemistry. Normalize the two functions and show they are orthogonal. 

2. Show that the first two wavefunctions of the harmonic oscillator (McQuarrie Table 5.3, p. 170) are normalized and that they are orthogonal to each other. a. Vo is normalized: Y(x) 8 114 _xx ² 1/2. e b. ₁ is normalized: 2 3 1/4 Y₁(x) = (40³) " Hox²) xe xx²/₂ c. o and ₁ are orthogonal:
Transcribed Image Text:2. Show that the first two wavefunctions of the harmonic oscillator (McQuarrie Table 5.3, p. 170) are normalized and that they are orthogonal to each other. a. Vo is normalized: Y(x) 8 114 _xx ² 1/2. e b. ₁ is normalized: 2 3 1/4 Y₁(x) = (40³) " Hox²) xe xx²/₂ c. o and ₁ are orthogonal:
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Can you explain how we get the circled terms please? 

a) Normalization condition is √" 1460)| ² dx = 1 or [ ^ √(x) 4(x) dx = 81 de LOHS = (+) [^*^ 46(21). 46(2) dr = √ ( =) * - ** ². (=) ¹¹ é Sa D 1/4 L 1 e (2) ²2500 (1 = 8 0+1 (2) ½ = (4x) x (12²) 12. 2x² "dx. 100 n=0 Here 2.(α)3/2 STT So Y() is Normalized. (16) [² 4³3 4,0x = (2-²) ³ x² = x (²) (x) dx 1/25 42² dx X 42³ 1/₂ √3/₂2 = 1² (x) 2월 2+1 (α)3/2 √T fousing I'm² e-ox² May = 1/2. 11/₂2 (+) 2 (0)3/2 x JTT = 1 but √2/2 = √FT { using fame anda = =0 so () is Normalized, (c) orthogonality condition f(x) 4₁ (2) dx = 0 0. 02² lites [ ^44) 4 (1) du = [(#)" (²) "^ 'n e (芋)(笑) (²) dn 14 - 81 х.е Sating upydneo Now √n+1 = nim 30 11/2/2 = 1/2+1 = 1/2 √2/2 = 1/2 √ using [² f(n) da = 0 if f(x) is add S La (a) n+1 Yo & 4, to each other. dr odd functio are orthogond JL
Transcribed Image Text:a) Normalization condition is √" 1460)| ² dx = 1 or [ ^ √(x) 4(x) dx = 81 de LOHS = (+) [^*^ 46(21). 46(2) dr = √ ( =) * - ** ². (=) ¹¹ é Sa D 1/4 L 1 e (2) ²2500 (1 = 8 0+1 (2) ½ = (4x) x (12²) 12. 2x² "dx. 100 n=0 Here 2.(α)3/2 STT So Y() is Normalized. (16) [² 4³3 4,0x = (2-²) ³ x² = x (²) (x) dx 1/25 42² dx X 42³ 1/₂ √3/₂2 = 1² (x) 2월 2+1 (α)3/2 √T fousing I'm² e-ox² May = 1/2. 11/₂2 (+) 2 (0)3/2 x JTT = 1 but √2/2 = √FT { using fame anda = =0 so () is Normalized, (c) orthogonality condition f(x) 4₁ (2) dx = 0 0. 02² lites [ ^44) 4 (1) du = [(#)" (²) "^ 'n e (芋)(笑) (²) dn 14 - 81 х.е Sating upydneo Now √n+1 = nim 30 11/2/2 = 1/2+1 = 1/2 √2/2 = 1/2 √ using [² f(n) da = 0 if f(x) is add S La (a) n+1 Yo & 4, to each other. dr odd functio are orthogond JL
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