Questions & Exercises on Heisenberg's Uncertainty Principle

1. In two-dimensional geometry we define a vector to be an entity that has both magnitude and direction. A typical vector, then, is drawn as a line segment with an arrowhead. These entities have their own rules of mathematics. For example, a vector can be added to another vector to generate a third vector, but according to the rules of "vector addition". This rule states that to add two vectors, say A and B, you have to place them head-to-tail by moving one vector parallel to itself (this operation does not change the vector). The new vector, C, which starts from the tail of one and ends at the head of the second one is the geometrical sum of the two vectors, i.e. C = A + B. This is illustrated to the right and is called the triangle rule of vector addition.  Using simple geometry, show that the parallelogram rule, which is illustrated as part ii, is equivalent to the triangle rule of vector addition.
    
2. A vector multiplied by a number (scalar) remains a vector of the same direction, but of a different length. We define a unit vector, i, to be a vector of length one along the +x-direction; and a unit vector, j, to be a vector of length one along the +y-direction. (There are our basis vectors). Use these rules to draw the vector:
    
   1. r = 3i + 5j
   2. s = i - 3j
   3. t = 2i + 3j
   4. What is the vector k = r + s ?
       
3. Use the mathematical relationship discussed in above two questions to prove that any arbitrary vector, say Q, could be written as a linear combination of the basis vectors, i and j.
    
4. Use a spreadsheet such as Excel to create the graph of a square wave by adding the first four Fourier components in its Fourier Series representation (see the mathematical relationship in the section: Fourier Series).
    
5. Calculate the frequency of oscillations of light coming out of the He-Ne laser that you used in question #5 of the last module (it is 632.8 nm!).
    
6. What is the wavelength of the radio waves that your FM receiver captures and turns into music?
    
7. The typical microwave oven transmits electromagnetic waves with a wavelength of few centimeters. What is the frequency of these waves?
    
8. Estimate the mass and the maximum speed of a baseball. (a) What is the maximum momentum of the baseball? (b) What is the deBroglie wavelength of the baseball? (c) Why is this value physically irrelevant? (The size of a proton/neutron is about 10-15 m.)
    
9. The mass of a hydrogen atom is about 1.67x10-27 kg. At room temperature it travels with a speed of about 2,700 m/s. Calculate its deBroglie wavelength. Is this value physically relevant? (Size of a typical atom is about 0.1 nm; that of simple molecules is about 1 nm.)
    
10. Does the Heisenburgh Uncertainty Principle state that we cannot determine the position of an object with unlimited accuracy? How about the momentum of the object? If not, then in what way does it set a limit on our knowledge?
     
11. Is Heisenburgh Uncertainty Principle a consequence of one measurement affecting the other; i.e. measuring the position accurately causes the velocity measurement to become inaccuate?