- Using The Result Of Example 3 8 In The Textbook Answer The Following Questions A Calculate The Energy Expectation Va 1 (21.96 KiB) Viewed 15 times
- Using The Result Of Example 3 8 In The Textbook Answer The Following Questions A Calculate The Energy Expectation Va 2 (21.96 KiB) Viewed 15 times
- Using The Result Of Example 3 8 In The Textbook Answer The Following Questions A Calculate The Energy Expectation Va 3 (21.96 KiB) Viewed 15 times
- Using The Result Of Example 3 8 In The Textbook Answer The Following Questions A Calculate The Energy Expectation Va 4 (91.45 KiB) Viewed 15 times
Using the result of Example 3.8 in the textbook, answer the following questions. (a) Calculate the energy expectation value, (E)(t) = (S(t)|Â|S(t)) and describe its change in time. - (b) Calculate the uncertainty of energy, AE(t) = √√(E²)(1) – ((E)(1))².
Using the result of Example 3.8 in the textbook, answer the following questions. (a) Calculate the energy expectation value, (E)(t) = (S(t)|Â|S(t)) and describe its change in time. - (b) Calculate the uncertainty of energy, AE(t) = √√(E²)(1) – ((E)(1))².
R3 Formalism Example 3.8 Imagine a system in which there are just two linearly independent states:32 and (2) = (1) = -(1) The most general state is a normalized linear combination: [S) = a11) + b12) = with a2 +1b2=1 The Hamiltonian can be expressed as a (hermitian) matrix (Equation 3.83); suppose it hau the specific form H₂ where g and h are real constants. If the system starts out (at = 0) in state 1), what is its state at time t? Solution: The (time-dependent) Schrödinger equation says ih (S(t)) = À\S(r)). (3.87) dt As always, we begin by solving the time-independent Schrödinger equation: Ĥs) = Els): (3.88) that is, we look for the eigenvectors and eigenvalues of Ĥ. The characteristic equation determines the eigenvalues: det E 8 =(-E)²-g²=0⇒h-E=g ⇒ E=htg. 8 Evidently the allowed energies are (h+g) and (hg). To determine the eigenvectors, we write = (hg) ⇒ha + gß = (hg)a⇒ B=ta, so the normalized eigenvectors are () Next we expand the initial state as a linear combination of eigenvectors of the Familtonian JS(0)) = ( 6 ) = √ √/12 (18+) + 18² 117 3.6 Vectors and Operators - Finally, we tack on the standard time-dependence (the wiggle factor) exp(-i Ent/h): 1 IS()) = e-(+)/(x+)+(-3)/x-)] MC/K HAN ()+ +²/l *(-)] e-/h +elatih) + clarsh) = e-the/h ( cos(f/h) ) sin(gr/h) If you doubt this result, by all means check it: Does it satisfy the time-dependent Schrödinger equation (Equation 3.87)? Does it match the initial state when I = 04 Just as vectors look different when expressed in different bases, so too do operators (or, in the discrete case, the matrices that represent them). We have already encountered a particularly nice example: (in position space). (the position operator) → ina/ap space): -iha/ax p (the momentum operator) →→→ (in position space). (in momentum space). ("Position space" is nothing but the position basis; "momentum space" is the momentum basis.) If someone asked you, "What is the operator, f, representing position, in quantum mechanics?" you would probably answer "Just x itself." But an equally correct reply would be "iha/ap." and the best response would be "With respect to what basis?" I have often said "the state of a system is represented by its wave function, (x, 1)," and this is true, in the same sense that an ordinary vector in three dimensions is "represented by the triplet of its components; but really, I should always add "in the position basis." After all, the state of the system is a vector in Hilbert space. (S(t)); it makes no reference to any particular basis. Its connection to (x,1) is given by Equation 3.77: V(x.r) = (x)S(t)). Having said. work in position space, and no serious harm comes from that, for the most part we do in referring to the wave function as "the state of the system." 3.6.2 Dirac Notation Dirac proposed to chop the bracket notation for the inner product, (a B), into two pieces, which he called ca, (a), and ket, (B) (I don't know what happened to the c). The latter is a vector, but whtly is the former? It's a linear function of vectors, in the sense that when it hits a vector (right) it yields a (complex) number the inner product. (When an operator hits a Một livers another vector: when a bra hits a vector, it delivers a number.) In a function space can be thought of as an instruction to integrate: (1= -fr f) dx.