12 - Physics 6210/Spring 2007/Lecture 12 Lecture 12...

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Physics 6210/Spring 2007/Lecture 12 Lecture 12 Relevant sections in text: § 2.1 The Hamiltonian and the Schr¨ odinger equation Consider time evolution from t to t + . We have U ( t + , t ) = I + - i ¯ h H ( t ) + O ( 2 ) . As usual, the unitarity of U implies that H ( t ) is Hermitian, i.e., it represents an observable – the Hamiltonian. There is one significant difference between the spatial momentum and the Hamiltonian, however. The spatial momentum is defined once and for all by its geometrical nature (in the Schr¨ odinger picture). The Hamiltonian depends on the details of the interactions within the system and with its environment. Thus there can be many useful Hamiltonians for, say, a particle moving in 3-d, but we always use the same momentum operator (in the Schr¨ odinger picture). Let us extract the Hamiltonian in a different way. We have that U ( t + , t 0 ) = U ( t + , t ) U ( t, t 0 ) . Substitute our result above for U ( t + , t ), divide both sides by and take the limit as 0, thereby defining the derivative of U . We get (exercise) i ¯ h dU ( t, t 0 ) dt = H ( t ) U ( t, t 0 ) . Let us turn the logic of this around. Let us suppose that, given a self-adjoint Hamiltonian, H ( t ), we can define U ( t, t 0 ) by the above differential equation.* When solving the differ- ential equation an initial condition will have to specified in order to get a unique solution. The initial condition we need is that U ( t 0 , t 0 ) = I. Thus we can say that, given a Hamiltonian, the time evolution of the system is determined. By focusing attention on H rather than U we get a considerable advantage in our ability to describe physical systems. Indeed, we shall always define the dynamics of a system by specifying its Hamiltonian. Note that it is much easier to give a formula for the energy of * This supposition can be proved rigorously when the Hamiltonian doesn’t depend upon time. One will have to make additional hypotheses in the more general case, but we won’t worry with those technical details. 1
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Physics 6210/Spring 2007/Lecture 12 a dynamical system than to explicitly display its dynamical behavior. Indeed, rarely will we be able to explicitly compute the time evolution operator. The relationship we just derived between the time evolution operator and the Hamil- tonian is an abstract version of the Schr¨ odinger equation. To see this, simply apply both sides of this operator relation to an arbitrary state vector, representing the initial state of a system at time t 0 . We have i ¯ h dU ( t, t 0 ) dt | ψ, t 0 = H ( t ) U ( t, t 0 ) | ψ, t 0 .
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