Page 684. Lecture 40: Coordinate Transformations: Time Transformations Date Revised: 2009/02/02 Date Given: 2009/02/02

Transcription

1 Page 684 Lecture 40: Coordinate Transformations: Time Transformations Date Revised: 2009/02/02 Date Given: 2009/02/02

2 Time Transformations Section 12.5 Symmetries: Time Transformations Page 685 Time Translation vs. Coordinate Transformations At a fundamental level, the time coordinate is different from the spatial coordinates in non-relativistic classical and quantum mechanics. The main reason is that time is considered a parameter: in classical mechanics, it is not a dynamical variable; in QM, there is no observable operator associated with time. There is no momentum (operator) conjugate to time with which to construct a Poisson bracket (commutation) relation. Energy is, to some extent, the observable conjugate to time, but it is only a rough correspondence between (t, H) and (X, P). Another problematic issue is that the Schrödinger Equation tells us how time evolution i.e., time translation should be done. We don t have any freedom in defining the time translation. Or, we could assume we could define time translations as we like, but they would have no physical relevance. For example, it would be nonsense to define a transformation T (t) q(t = 0) = q = q(t) because we know that, even for an explicitly time-independent Hamiltonian, q is in general not an eigenstate of H and so its time evolution is not a simple time-dependent mapping to other position-basis elements in the space.

3 Section 12.5 Symmetries: Time Transformations Page 686 That said, we can see that much of our formalism for coordinate transformations can be carried over to time translations. Let s define time translation to be the standard unitary time-evolution operator, but allowing for the fact that H may be time-dependent and may not commute with itself at different times ([H(t), H(t )] = 0 is only guaranteed for t = t ). That is, T (t) = U(t) = T» exp i Z t 0 «N 1 Y» dt H(t ) = lim exp i ««t j H N N N t j=0 (Equation 4.19) That is, there is a very specific transformation that provides time translation: the only freedom is in the amount of time t. The form of the translation depends on the particular Hamiltonian, unlike any of the coordinate transformations, which can be applied to any system with any Hamiltonian. We know the above transformation is unitary from previous work. We will write the transformation as ψ(t) = ψ = U(t) ψ(t = 0)

4 Section 12.5 Symmetries: Time Transformations Page 687 For an arbitrary operator, we take as a requirement ψ 1 (t) O(t) ψ 2 (t) ψ 1 O ψ 2 = ψ 1(t = 0) O 0 ψ 2 (t = 0) O(t) O = U(t) O 0 U (t) where we define our untransformed operator to be O 0 to avoid confusion between O 0 and O(t) (i.e., if we had taken our untransformed operator to be just O, as we did for coordinate transformations, it would be unclear whether O refers to O(t) or O(t = 0)). Of course, O 0 = O(t = 0). In addition to the above relation between the matrix elements of O(t) in the transformed basis and the matrix elements of O 0 in the untransformed basis, we might be inclined to ask whether there is a relation between the matrix elements of O(t) in the untransformed basis and the matrix elements of O 0 in the transformed basis. As we frequently find, there is no such relation in general: ψ 1 (t) O 0 ψ 2 (t) = ψ 1 (t = 0) U (t) O 0 U(t) ψ 2 (t = 0) ψ 1 (t = 0) O(t) ψ 2 (t = 0) = ψ 1 (t = 0) U(t) O 0 U (t) ψ 2 (t = 0) The expressions simplify and are equal if [O 0, U(t)] = 0 for all t, which is equivalent to [O 0, H(t)] = 0 for all t. But, in that case, O(t) = O 0 and O is conserved.

5 Section 12.5 Symmetries: Time Transformations Page 688 Even for a time-dependent Hamiltonian, it holds that the infinitesimal time evolution operator that goes from t to t + dt is U(t + dt, t) = 1 i H(t) dt If H is time-dependent, knowing that H is the generator is not particuarly helpful because there is no fixed basis of eigenstates of H. So, at this point, let us specialize to time-independent H. The time translation operator has an explicit form, U(t) = e i H t As note above, the usual coordinate transformation formula, q = T q does not hold because T q = e i H t q(t = 0) is in general not a position basis element. The other two general relations do carry through: ψ ψ(t) = e i H t ψ(t = 0) e i H t ψ O(t) O = e i H t O e i H t

6 Section 12.5 Symmetries: Time Transformations Page 689 As we did for coordinate transformations, we are led to consider the effect of the transformation on the eigenstates of the generator: E(t) E = e i H t E = e i E t E We may also follow the example from coordinate transformations for transformation of the generator-eigenbasis representation of an arbitrary state ψ. That is, consider the transformation of an arbitrary state ψ when written in the { E }-basis representation. We denote this representation, E ψ(t) E ψ, by ψ E (E, t) ψ E (E), and term it the { E }-basis wavefunction or energy-basis wavefunction (it is not to be confused with our notation ψ E for the Hamiltonian eigenstate with energy E). It is E ψ(t) E ψ = E e i H t ψ = e i E t E ψ = e i E t ψ E (E)

7 Section 12.5 Symmetries: Time Transformations Page 690 Consider the time transformation of matrix elements of the operators in the energy eigenbasis: E 1 O(t) E 2 = E 1 O E 2 = E 1 e i H t O e i H t E 2 = e i (E 1 E 2 ) t E 1 O E 2 So, in spite of the imperfect analogy between the time-translation transformation and coordinate transformations, we see that many of the general results for coordinate transformations carry through for time translation. As a final note, we state the obvious: the generator H commutes with the Hamiltonian H, so the eigenstates of the Hamiltonian are eigenstates of the generator and vice versa, and the generator H is conserved in all the usual ways.

8 Section 12.5 Symmetries: Time Transformations Page 691 Time Reversal Transformation The other analogy between time and coordinate transformations that we should consider is the temporal equivalent of a parity transformation, the time reversal transformation. As with translations, there are some subtleties that arise. One might think that the right rule for the transformation would be Π t ψ(t) = ψ( t) Let s see what time evolution equations ψ( t) satisfies. To avoid confusion, define φ(t) = ψ( t) and work on it, changing variables to t = t and then back to t to make the manipulations clearer: i d dt φ(t) = i d d ψ( t) = i dt ψ( t) = i d d( t) dt ψ(t ) = H(t ) ψ(t ) = H( t) ψ( t) = H( t) φ(t)

9 Section 12.5 Symmetries: Time Transformations Page 692 So, φ(t) = ψ( t) satisfies the Schrödinger Equation with a Hamiltonian that is the negative of the time reverse of the Hamiltonian for which ψ(t) satisfies the Schrödinger Equation. This is clearly not useful if we consider an eigenstate of a time-independent H: if ψ(t) = E(t) is an eigenstate of H with energy E, then ψ( t) is an eigenstate of H with energy E! If we look at the Schrödinger Equation in the position basis, we are led to a slightly different definition of the operation of the parity operator. Suppose that ψ(t) is a solution of the Schrödinger Equation with Hamiltonian H. Then the position-basis Schrödinger Equation is i d dt ψx (x, t) = H x, i d dx, t «ψ x (x, t) where we have shown H as a function of x and i d instead of X and P because dx we are working in the position basis. Take the complex conjugate: i d dt ψ x (x, t) = H x, i d dx, t «ψ x (x, t)

10 Section 12.5 Symmetries: Time Transformations Page 693 Now, define φ x (x, t) = ψ x (x, t). Then we have i d dt φx (x, t) = H x, i d dx, t «φ x (x, t) Change variables to t = t: i d d( t ) φx (x, t ) = H x, i ddx «, t φ x (x, t ) i d dt φx (x, t ) = H x, i ddx «, t φ x (x, t ) So we see that φ x (x, t) is the position-space representation of a state that satisfies the Schrödinger Equation for the Hamiltonian that is the time-reversed, complex conjugate of the original Hamiltonian. This is a much more reasonable result than what we had before.

11 Section 12.5 Symmetries: Time Transformations Page 694 So, we define Π t ψ(t) = φ(t) with φ x (x, t) = ψ x (x, t) x φ(t) = ψ( t) x The second part of the definition, x φ(t) = ψ( t) x, corrects what was wrong with the first definition, which would have defined x φ(t) = x ψ( t). The subtlety here is that, while complex conjugation is perfectly well defined for the ψ x (x, t), it is not defined for a Hilbert space vector ψ( t), so we had to go through the position-basis representation of the Schrödinger Equation to see that the complex conjugation step was needed. We see from the above that the condition for time reversal to be a symmetry of the Hamiltonian is for H = H ; if this happens, then the Schrödinger Equation that φ(t) satisfies in the position basis is the same as the one that ψ(t) satisfies. It would be nice to be able to write this last statement (and, in general, transformation rules for operators under Π t) in a form similar to what we did for the spatial parity transformation, but the lack of a direct ket-to-ket mapping makes this difficult.