* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download ANGULAR MOMENTUM IN QUANTUM MECHANICS
Noether's theorem wikipedia , lookup
Renormalization wikipedia , lookup
Quantum group wikipedia , lookup
Double-slit experiment wikipedia , lookup
Identical particles wikipedia , lookup
Wave function wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Quantum key distribution wikipedia , lookup
History of quantum field theory wikipedia , lookup
Coherent states wikipedia , lookup
Bell's theorem wikipedia , lookup
Probability amplitude wikipedia , lookup
Atomic orbital wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Path integral formulation wikipedia , lookup
Renormalization group wikipedia , lookup
Quantum entanglement wikipedia , lookup
Quantum teleportation wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Spin (physics) wikipedia , lookup
Wave–particle duality wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Canonical quantization wikipedia , lookup
Matter wave wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Quantum state wikipedia , lookup
Hidden variable theory wikipedia , lookup
EPR paradox wikipedia , lookup
Particle in a box wikipedia , lookup
Hydrogen atom wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
ANGULAR MOMENTUM IN QUANTUM MECHANICS I. Orbital angular momentum A. Consider a particle in a three-dimensional potential that has quantum number l = 3. 1. Suppose you were to measure the magnitude of the orbital angular momentum vector for this particle. What value(s) could you get? Explain. 2. Can the magnitude of the orbital angular momentum vector for a particle be negative? Can the quantum number l for a particle be negative? The orbital angular momentum vector satisfies the following eigenvalue equation: L̂2 l, ml = l(l +1) 2 l, ml where l is a nonnegative integer. 3. Resolve any inconsistencies between this expression and your answers above. ! ! B. Consider a classical vector A with magnitude 3.5. The direction of A is not known. ! 1. Determine the largest possible value for the z-component of A . ! 2. Determine the smallest possible value for the z-component of A . (Hint: Can a single component of a vector be negative?) ! 3. Determine the set of all possible values for the z-component of A . Explain. Tutorials in Physics: Quantum Mechanics ©McDermott, Heron, Shaffer, and P.E.G., U. Wash. Preliminary First Edition, 2014 QM 1 QM Angular momentum in quantum mechanics 2 C. Suppose you were to measure the magnitude of the z-component of the orbital angular momentum vector for a particle that has l = 3. Determine the set of all possible value(s) for this measurement. Explain. (Hint: Use your argument from part B, but recall the quantization condition for angular momentum in quantum mechanics.) The z-component of the orbital angular momentum satisfies the following eigenvalue equation: Lˆ z l , ml z = ml ! l , ml z where ml is an integer with ml ≤ l. D. Determine the set of all possible values that could result from a measurement of the x-component of the particle’s angular momentum. Assume no other measurements have been made. Explain. (Hint: Is there anything unique about the z-direction?) Would your answer change if you were to consider a measurement of the component of angular momentum along an arbitrary axis (e.g., the y-axis, or halfway between the x- and z-axes)? E. Do you agree or disagree with the following statement? Explain. ”When we measure the z-component of L, we always get something less than the total L, so angular momentum can never point in the z-direction. But if we happened to choose the direction in which L did point, then we could measure the full magnitude.” F. Consider a particle for which only the quantum number ml is known. What are the possible values of the quantum number l for this particle? Explain. ! Check your results with a tutorial instructor. Tutorials in Physics: Quantum Mechanics ©McDermott, Heron, Shaffer, and P.E.G., U. Wash. Preliminary First Edition, 2014 Angular momentum in quantum mechanics QM 5 III. Angular momentum and uncertainty Consider a particle with the initial orbital angular momentum state ψ = l, ml z = 2,1 z . A. Determine two quantities related to orbital angular momentum that are well-defined for this particle, and the value for each. Explain. B. Is the x-component of the orbital angular momentum, Lx, well-defined for the initial state given above? Explain. If Lx is well-defined, what is its value? If it is not well-defined, determine the possible results of a measurement of the x-component of the orbital angular momentum. Explain. What additional information would you need to determine the probabilities of the possible results of this measurement? Explain. C. Suppose instead that you were to measure the y-component of the orbital angular momentum, Ly. Would your answers to part B change? Explain. Suppose that a measurement of the y-component of the orbital angular momentum of the initial state above resulted in +ħ. D. Consider the student discussion below. Student 1: ”We started knowing the z-direction was 1ħ, and now we know the y-direction is also 1ħ. The magnitude of angular momentum is only 2ħ, so the x-component must be 0.” Student 2: ”Angular momentum squared is l(l +1), which for this state is 6ħ2. I know that L2 = L2x + L2y + L2z , so if y and z are both 1ħ, I get that the x-component must be 2ħ.” Neither statement is completely correct. Explain where each statement is incorrect. Tutorials in Physics: Quantum Mechanics ©McDermott, Heron, Shaffer, and P.E.G., U. Wash. Preliminary First Edition, 2014 QM Angular momentum in quantum mechanics 6 Suppose that a measurement in the x-direction resulted in -2ħ; e.g., ψ = l, ml x = 2, −2 x . E. Suppose you made a second measurement, this time of the z-component. 1. Predict the ranking of the probabilities of all the possible results of this measurement from most probable to least probable. Explain your reasoning. The angular momentum state of this particle can be rewritten in terms of the z-basis, as follows: 2, −2 x = 1 2, 2 z − 2 2,1 z + 6 2, 0 z − 2 2, −1 z + 2, −2 4 ( z ). 2. Use the expression above to determine the probability associated with each value of Lz for this particle. Show your work. Resolve any inconsistencies with your prediction. All three components of a quantum mechanical angular momentum vector cannot be welldefined at any given instant; however, classical reasoning can still help make predictions about the probabilities associated with the allowed values of an angular momentum measurement. F. Use your knowledge of classical vectors to account for each of the following pieces of information about the particle above: 1. The most likely result of a measurement of Lz is 0. 2. The probabilities are the same for positive and negative results with the same absolute value. 3. The probabilities are not equally distributed among the possible values. According to the uncertainty principle, only one of the components of angular momentum in quantum mechanics may be known precisely at any one instant. If a particle is in an eigenstate of L̂z then a measurement along a perpendicular axis, such as L̂x , can (usually) result in any allowed eigenvalue of L̂x . You will investigate this further in the homework. Tutorials in Physics: Quantum Mechanics ©McDermott, Heron, Shaffer, and P.E.G., U. Wash. Preliminary First Edition, 2014