CHEM 442 Lecture 3 Problems 3-1. List the similarities and differences between the classical- and quantum-mechanical equations of motion. How does the latter account for the quantization of energy and wave-particle duality? 3-2. Suggest mathematical functions that represent a sinusoidal wave of amplitude A, frequency n , and wavelength l . 3-3. Justify the mathematical form of the linear momentum operator, -i d , by applying dx it to the simple sinusoidal wave of 3-2. 3-4. Justify the mathematical form of the energy operator, i d , by applying it to the dt simple sinusoidal wave of 3-2. 3-5. Write down the time-independent and time-dependent Schrödinger equations of a 1 particle with mass m confined in the one-dimensional potential kx 2 , where k is some 2 constant. 3-6. What is an eigenvalue equation? 3-7. Starting with Hamilton’s equation of motion for classical mechanics, obtain Newton’s equation of motion, F = ma . p2 +V = E, 2m d2 d + 2 + 3x̂ , on the following functions: (a) 2x , (b) eax , (c) 3-8. Act the operator, 2 dx dx . sin kx 3-9. An operator Ŵ is said to be linear if Ŵ éë c1 f1 ( x ) + c2 f2 ( x ) ùû = c1Ŵf1 ( x ) + c2Ŵf2 ( x ) . In quantum mechanics, all operators are linear. Determine whether the following operators d2 d 3 x 2 , (d) ( ) , (e) ( ) . In the last two, ( ) are linear or nonlinear: (a) , (b) 2 , (c) dx dx is where the function acted on by the operator goes.