Griffith Intro to QM, 1st order perturb theory in time-dep

• tjsgkdms1111
In summary, the conversation discusses the topic of time-dependent perturbation theory and its application to finding the coefficients of the wave function. The main result is that at first order, the sum of the squared coefficients may be greater than one, which can be resolved by going to higher order or modifying the coefficient for conservation of the norm.
tjsgkdms1111

Homework Statement

ca(0)=1, cb(0)=0
zeroth: ca(t)=1, cb(t)=0
1st: ca(t)=1, cb(t)=i/hbar*integral(H'(t) exp(iwt)) dt
ca^2+cb^2=1 to 1st order of H'.: What does it mean?
it is evidently not 1 at all.

The Attempt at a Solution

tjsgkdms1111 said:

Homework Statement

ca(0)=1, cb(0)=0
zeroth: ca(t)=1, cb(t)=0
1st: ca(t)=1, cb(t)=i/hbar*integral(H'(t) exp(iwt)) dt
ca^2+cb^2=1 to 1st order of H'.: What does it mean?
it is evidently not 1 at all.

The Attempt at a Solution

Is this really a homework question, or you don't understand something about TDPT?

While your question is not clear, I get the feeling that you are bothered by the fact that ##|c_a|^2 + |c_b|^2 > 1##, but this is indeed the result you get from 1st-order TDPT. You have to go to higher order to resolve this, or take that by conservation of the norm ##c_a## must be modified such that ##|c_a|^2 = 1 - |c_b|^2 ## (but that only tells you the probability of staying gin the initial state, and not the actual complex coefficient ##c_a##).

1. What is the Griffith Intro to QM course about?

The Griffith Intro to QM course is an introductory course in quantum mechanics that covers the basic principles and mathematical framework of the field. It also includes an introduction to first-order perturbation theory in time-dependent systems.

2. What is perturbation theory in quantum mechanics?

Perturbation theory in quantum mechanics is a mathematical tool used to approximate solutions to complex quantum systems by adding a small perturbation to a known, solvable system. It is particularly useful for understanding the behavior of systems that are subject to external influences or time-dependent changes.

3. What is first-order perturbation theory in time-dependent systems?

First-order perturbation theory in time-dependent systems is a specific application of perturbation theory that allows for the calculation of the effects of a small, time-dependent perturbation on a quantum system. It involves expanding the time-dependent wave function in terms of the unperturbed wave functions and using time-dependent perturbation theory to calculate the coefficients of the expansion.

4. Why is perturbation theory important in quantum mechanics?

Perturbation theory is important in quantum mechanics because it allows for the calculation of approximate solutions to complex systems that would otherwise be impossible to solve exactly. It is also a useful tool for understanding the effects of external influences or time-dependent changes on quantum systems.

5. How can I apply first-order perturbation theory in time-dependent systems to real-world problems?

First-order perturbation theory in time-dependent systems can be applied to a wide range of real-world problems, such as the behavior of atoms and molecules in electromagnetic fields, the interaction between light and matter, and the dynamics of chemical reactions. It is also used in fields such as solid-state physics, nuclear physics, and particle physics to understand the behavior of quantum systems.

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