Time Dependent Perturbation Theory
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The presentation is about how to evaluate the probability of finding the system in any particular state at any later time when the simple Hamiltonian was added by time dependent perturbation. So now the wave function will have perturbation-induced time dependence.
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The presentation is about how to solve the new energy levels and wave functions when the simple Hamiltonian is added by another term due to external effect (can be due to external field) .
The intended reader of this presentation were physics students. The author already assumed that the reader knows dirac braket notation.
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Perturbation theory allows approximations of quantum systems where exact solutions cannot be easily determined. It involves splitting the Hamiltonian into known and perturbative terms. For the helium atom, the zero-order approximation treats it as two independent hydrogen atoms, yielding the wrong energy. The first-order approximation includes repulsion between electrons, giving a better but still incorrect energy. Variational theory provides an energy always greater than or equal to the actual energy.
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Perturbation theory provides an approximate method for solving quantum mechanical problems where the Hamiltonian cannot be solved exactly. It involves splitting the Hamiltonian into an exactly solvable unperturbed part (H0) and a perturbed part (H1) treated as a small disturbance. The eigenvalues and eigenstates of the full Hamiltonian are expressed as power series expansions in terms of the perturbation strength parameter λ, allowing the effects of the perturbation to be calculated order by order. There are two types of perturbation theory: time-independent, where the unperturbed eigenstates are stationary; and time-dependent, where they vary with time under the perturbation.
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The presentation is about how to solve the new energy levels and wave functions when the simple Hamiltonian is added by another term due to external effect (can be due to external field) .
The intended reader of this presentation were physics students. The author already assumed that the reader knows dirac braket notation.
Brief introduction to perturbation theory
Perturbation theory allows approximations of quantum systems where exact solutions cannot be easily determined. It involves splitting the Hamiltonian into known and perturbative terms. For the helium atom, the zero-order approximation treats it as two independent hydrogen atoms, yielding the wrong energy. The first-order approximation includes repulsion between electrons, giving a better but still incorrect energy. Variational theory provides an energy always greater than or equal to the actual energy.
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Perturbation theory provides an approximate method for solving quantum mechanical problems where the Hamiltonian cannot be solved exactly. It involves splitting the Hamiltonian into an exactly solvable unperturbed part (H0) and a perturbed part (H1) treated as a small disturbance. The eigenvalues and eigenstates of the full Hamiltonian are expressed as power series expansions in terms of the perturbation strength parameter λ, allowing the effects of the perturbation to be calculated order by order. There are two types of perturbation theory: time-independent, where the unperturbed eigenstates are stationary; and time-dependent, where they vary with time under the perturbation.
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This document discusses time-dependent perturbation theory. It begins by introducing the concept of applying a time-dependent perturbation to a quantum system to induce transitions between its energy eigenstates. It then describes how the interaction picture can be used to focus on the slow evolution induced by the perturbation, without considering the rapid oscillation from the unperturbed Hamiltonian. The interaction picture defines a transformed state vector and operators such that the perturbation Hamiltonian governs the evolution operator in a Schrodinger equation.
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Time Dependent Perturbation Theory
- 1. Time-Dependent Perturbation Theory Prepared by: James Salveo L. Olarve Graduate Student January 28, 2010
- 2. Introduction The presentation is about how to evaluate the probability of finding the system in any particular state at any later time when the simple Hamiltonian was added by time dependent perturbation. S o now the wave function will have perturbation-induced time dependence. The intended reader of this presentation were physics students. The author already assumed that the reader knows dirac braket notation. This presentation was made to facilitate learning in quantum mechanics.
- 3. Time-Dependent Perturb Hamiltonian We look at a Hamiltonian is some time-dependent perturbation in which we assumed to be small compared to the time-independent part. here unperturbed eigenvalue equation Note: We label here not since for a time-dependent Hamiltonian, the energy will not be conserved. Therefore energy corrections are futile to solve Now, even for V=0, the wave functions have the usual time dependence with as constant
- 4. On introducing the acquire time dependence Time-Dependent Perturb Hamiltonian This time dependence can be determined by Schödinger’s Equation with
- 5. Time-Dependent Perturb Hamiltonian Now, Taking the inner product with the bra and introducing
- 6. Time-Dependent Perturb Hamiltonian This is a matrix differential equation for the and solving this set of coupled equations will give us the , and hence the probability of finding the system in any particular state at any later time.
- 7. If the system is in initial state at t = 0, the probability amplitude for it being in state at time t is to leading order in the perturbation Time-Dependent Perturb Hamiltonian The probability that the system is in fact in state at time t is therefore Obviously, this is only going to be a good approximation if it predicts that the probability of transition is small—otherwise we need to go to higher order, using the Interaction Representation