# Sol2 - Assignment 2 1 Question 1(x = 1/4 x2/2 e Before...

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Assignment 2 1Q u e s t i o n 1 We’re given ψ ( x )= ± α π ² 1 / 4 e αx 2 / 2 . Before proceeding let’s check that this is normalized correctly since we’ll need that integral later on anyway. Consider some integral of the form I = Z + −∞ dxe ax 2 where a is real and positive. The function e ax 2 is called a Gaussian and this integral is extremely important. Integrals of this form come up all the time in physics, especially in statistical mechanics and quantum Feld theory. The square of I is I 2 = ³ Z + −∞ dxe ax 2 ´³ Z + −∞ dye ay 2 ´ = Z dxdye a ( x 2 + y 2 ) . This suggests changing variables in the integrand as x = r cos θ , y = r sin θ so that I 2 = Z 2 π 0 Z 0 re ar 2 dr which can be completed as I 2 =2 π " e ar 2 2 a # 0 = π a so I = r π a . or Z + −∞ e ax 2 dx = r π a (1) In the case at hand the normalization integral is Z dxψ ( x ) ? ψ ( x )=1 Z + −∞ r α π e αx 2 =1 Using (1) it’s trivial to see that ψ ( x ) is normalized correctly. 1

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What about the moments? It should be obvious that Z + −∞ xe ax 2 dx =0 (2) because this is an odd function integrated over even limits. (Similarly for x 3 ,x 5 and so on.) What about integrals of the form R + −∞ x 2 n e ax 2 dx where n is an integer? These can be performed with the help of a little trick: take the derivative of both sides of (1) with respect to a .Ige t Z + −∞ ( x 2 ) e ax 2 dx = 1 2 πa 3 / 2 so that Z + −∞ x 2 e ax 2 dx = 1 2 r π a 3 . (3) Youcancomputea l ltheevenmoments R + −∞ x 2 n e αx 2 dx in this way, by suc- cessively taking derivatives with respect to a . Now, on with the question at hand. We want to compute the expectation value of the momentum h ˆ p i = Z + −∞ ψ ( x ) ?
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Sol2 - Assignment 2 1 Question 1(x = 1/4 x2/2 e Before...

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