For a quantum particle in a cube, the total translational energy is given by E_n = (h^2/(8ma^2)) x [n_x^2 + n_y^2 + n_z^2], where the n_i are the translational quantum states in the x, y, and z dimensions, and a is the length/width/height of the box. What is the degeneracy when n_x^2 + n_y^2 + n_z^2 = 6?
The degeneracy can be thought of as the multiple different ways to satisfy the equation n_x^2 + n_y^2 + n_z^2 = 4, with different values of the n_i. This will tell us the different combination of eigenstates that sum to the same total energy of 6. Note that the values of the n_i are all positive integers (1, 2, 3, . . .) for a particle in a box. For this simple case, we simply enumerate the possibilities. We have (1)^2 + (1)^2 + (2)^2 = 6 [That is, n_x = 1, n_y = 1, n_z = 2] (1)^2 + (2)^2 + (1)^2 = 6 (2)^2 + (1)^2 + (1)^2 = 6 So the degeneracy is 3. For n_x = n_y = n_z = 1, the degeneracy would (obviously) be simply 1, because that is the only way for the sum of their squares to equal 3, when n is greater than or equal to 1.
A chemical element has 11 protons, 10 neutrons, and 10 electrons. What is the identity of this element?
An element is defined by the number of protons. Therefore, this must be some isotope of sodium, Na, according to the . Because there is one fewer electron than the number of protons, this is a sodium cation with a charge of +1. The mass number includes the number of protons plus neutrons, which is 11 + 10 = 21. The species is therefore 21Na+.
For the function f(x) = 2x^2 + 3x, determine the slope of the tangent line to f(x) at x = 4.
The tangent line is simply "math speak" for a line with an instantaneous slope equal to the derivative of f(x) at x = 4. The derivative f'(x) = 4x + 3 by the power rule, so f'(4) = 4(4) + 3 = 19. The tangent line has the equation f(x) - f(4) = 19(x - 4). Since f(4) = 2(4)^2 + 3(4) = 44, the tangent line has the equation f(x) - 44 = 19(x - 4). In slope-intercept form this is f(x) = 19x - 32.