# adiabatic - Phys 852 Quantum mechanics II Spring 2008 The...

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Phys 852, Quantum mechanics II, Spring 2008 The Quantum Adiabatic Theorem 4/14/2008 Prof. Michael G. Moore, Michigan State University 1 Statement of the Problem: The physics of adiabatic transfer is a common thread found throughout modern quantum-mechanics ex- periments. In a typical quantum system, the Hamiltonian contains one or more adjustable parameters. Thus it is of great practical importance to understand what happens to the state of a quantum system if these variables are varied with time. In general, there are two important limiting cases: (i) the parameter is varied much faster than the system response time, and (ii) the parameter is adjusted much slower than the system response time. This second limiting case is the adiabatic regime. Before analyzing these two limiting cases, we need to consider the structure of a time varying Hamiltonian. The eigenvalues of a Hamiltonian satisfy the energy eigenvalue equation H | E n ) = E n | E n ) . (1) If H H ( t ), then at fixed t , H is still a Hermitian matrix, and thus has a set of real eigenvalues and orthonormal eigenvectors. When t is varied, the local eigenstates and eigenvalues also change with time, so that the eigenvalue equation becomes H ( t ) | E n ( t ) ) = E n ( t ) | E n ( t ) ) . (2) The eigenstates satisfy the usual orthonormality condition at equal times ( E m ( t ) | E n ( t ) ) = δ mn , (3) but at different times the inner product ( E m ( t ) | E n ( t ) ) has no special significance. With this structure we can now consider the response of the system to a sudden (i) or adiabatic (ii) change in H . For simplicity we will consider the case where the system is prepared initially in the n th enegy level, so | ψ (0) ) = | E n (0) ) . . We now evolve time from t = 0 to t = T , with T being very small, yet H ( T ) being very different from H (0). In this regime, the state cannot respond, so it remains in its initial state, so that | ψ ( T ) ) = | E n (0) ) . But | E n (0) ) is no longer an eigenstate of H ( T ). To see how the state then evolves we must expand onto the new eigenstates, giving | ψ ( t ) ) = summationdisplay m | E m ( T ) )( E m ( T ) | E n (0) ) e m ( t T ) . (4) Because the expansion coefficients ( E m ( T ) | E n (0) ) are essentially random, we see that a sudden change takes a system in the n th eigenstate into a random superposition of many eigenstates. Hence we can say that a fast parameter change induces transitions between the local eigenstates. In the limit of a very slow change, we can invoke the Aidabatic theorem, which says that a system initially in state | E n (0) ) will be found with unit probability in state | E n ( T ) ) , i.e. no transitions have occured. We note that the state | E n ( T ) ) can be very different from the state | E n (0) ) . Their relationship is that they are both the n th eigenstate of their respective Hamiltonian.

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