2009_1_sol

# 2009_1_sol - (b Physically what does the wave function in...

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Name: Student Number: Quiz 1 ELEC 4705 Tuesday Sept. 29 2009 1. (5 marks) Operators. (a) Given a potential energy: V ( x, y ) = 5 x 4 + sin ( xy ) (1) and the operator = ˆ i ∂x + ˆ j ∂y (2) determine the gradient V as a function of x and y . ∂V ∂x = 20 x 3 + cos ( xy ) y ∂V ∂y = cos ( xy ) x V = (20 x 3 + cos ( xy ) y ) ˆ i + ( cos ( xy ) x ) ˆ j 2.5 marks (b) What does this gradient physically represent? Is it a vector or a scalar? The gradient represents the slope of the function – or the ﬁrst derivative 1.5 marks It is a vector and includes the direction of the slope. 1.0 mark

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ELEC 4705 Electronic Properties of Materials Quiz 1 2. Schrodinger equation (5 Marks) (a) If the potential in Question 1 was for an electron write down the 2 dimensional Schrodinger’s equation. 3 Marks - ¯ h 2 2 m 2 ψ ( r ) + V ( r )Ψ( r ) = E ψ ( r ) (3) - ¯ h 2 2 m ( 2 V ∂x 2 + 2 V ∂y 2 ) ψ ( x, y ) + (5 x 4 + sin ( xy )) ψ ( x, y ) = E ψ ( x, y ) (4)
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Unformatted text preview: (b) Physically what does the wave function in part (a) represent? • The probability of ﬁnding an electron at the point x . 2 marks 2 Sept. 29, 2009 ELEC 4705 Electronic Properties of Materials Quiz 1 3. Wave Packets (5 Marks) (a) Why do we represent an electron by a wave packet? • Because the free electron solution for a single phase is un-localized. (b) If ψ is deﬁned with respect to x (space) as: ψ ( x ) = e-( x-x ) 2 / 2(Δ x ) 2 (5) Sketch the wave function in the space. Gaussian distribution centered at x and with a width δx . What would be it’s functionality in k space? Sketch it as well. Gaussian distribution centered at k and with a width δk . 3 Sept. 29, 2009...
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2009_1_sol - (b Physically what does the wave function in...

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