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1. Evaluate g = Âf, where  and f are given below: (McQuarrie 3-1) f SQRT (V) (а) (b) x4 + x3 e ax dx' (c) dx x - 2x + 3 (d) Əx2

1. Evaluate g = Âf, where  and f are given below: (McQuarrie 3-1) f SQRT (V) (а) (b) x4 + x3 e ax dx' (c) dx x - 2x + 3 (d) Əx2

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1. Evaluate g = Âf, where  and f are given below: (McQuarrie 3-1)  SQRT (V) f (a) (b) x4 + x3 dx' eax (c) dx x - 2x + 3 (d) Əx? * ay2 2. Determine whether the following operators are linear or nonlinear: (McQuarrie 3-2) a. Āf(x) = SQRF(x) [square f(x)] b. Âf(x) = f'(x) c. Âf(x) = 0 d. Âf(x) = [f(x)]*1 e. Âf(x) = f(0) Ât(x) = Inf(x) [form the complex conjugate of f(x)] [multiply f(x) by zero] [take the reciprocal of f(x)] [evaluate f(x) at x=0] [take the logarithm of f(x)] f. 3. In each case, show that f (x) is an eigenfunction of the operator given. Find the eigenvalue. (McQuarrie 3-3) f(x) (а) d? coSWX dx2 d (b) e lwt dt (c) eax + 3 (d) ay 4. Determine whether or not the following pairs of operators commute. (McQuarrie 3-6) (a) d. + dx2 2 dx (b) X dx. SQRT (c) (d) SQR Əx ду 5. Show that <X> for all states of a particle in a box. Is this result physically reasonable? (McQuarrie 3-15) 6. Calculate <x> and <x?> for the n = 2 state of a particle in a one-dimensional box of length a. Show that: (McQuarrie 3-26)
Transcribed Image Text:1. Evaluate g = Âf, where  and f are given below: (McQuarrie 3-1)  SQRT (V) f (a) (b) x4 + x3 dx' eax (c) dx x - 2x + 3 (d) Əx? * ay2 2. Determine whether the following operators are linear or nonlinear: (McQuarrie 3-2) a. Āf(x) = SQRF(x) [square f(x)] b. Âf(x) = f'(x) c. Âf(x) = 0 d. Âf(x) = [f(x)]*1 e. Âf(x) = f(0) Ât(x) = Inf(x) [form the complex conjugate of f(x)] [multiply f(x) by zero] [take the reciprocal of f(x)] [evaluate f(x) at x=0] [take the logarithm of f(x)] f. 3. In each case, show that f (x) is an eigenfunction of the operator given. Find the eigenvalue. (McQuarrie 3-3) f(x) (а) d? coSWX dx2 d (b) e lwt dt (c) eax + 3 (d) ay 4. Determine whether or not the following pairs of operators commute. (McQuarrie 3-6) (a) d. + dx2 2 dx (b) X dx. SQRT (c) (d) SQR Əx ду 5. Show that <X> for all states of a particle in a box. Is this result physically reasonable? (McQuarrie 3-15) 6. Calculate <x> and <x?> for the n = 2 state of a particle in a one-dimensional box of length a. Show that: (McQuarrie 3-26)
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