Probability Amplitude and Time Evolution Operator

[erkokite]

Hello,

I am fairly new to quantum physics (I'm actually an engineer, not a physicist). I think I am getting a decent grasp on things, but I have a question.

Suppose you have two time dependent states: $$\Psi_{1}$$ and $$\Psi_{2}$$.

Also, suppose that we have a constant potential, represented in our Hamiltonian as V.

Our Hamiltonian is thus represented as: $$\hat{H}=1/(2m)\partial^2/x^2+V(x)$$

Now suppose that we want to find the probability amplitude to go from state 1 to the 2nd state in a time interval t1 to t2.

It is my understanding that the following operation is used:

$$<\Psi_{2}|exp(-i\hat{H}(t2-t1))|\Psi_{1}>$$

This is of course equal to

$$\int\Psi_{2}*exp(-i\hat{H}(t2-t1))\Psi_{1}dx$$

However, how do I evaluate the following operation?

$$exp(\hat{H})\Psi_{1}$$

Many thanks.

 

[nicksauce]

The general idea of evaluating a function of an operator is to expand the function...

exp(H) = 1 + H + H^2/2 + H^3/6 + ...

If Psi is an eigenfunction of H with eigenvalue E, then it is not hard to show that exp(H) Psi = exp(E) Psi.

 

[Entropia]

Hello,

To answer your question, we first need to understand the concept of the time evolution operator. The time evolution operator, denoted as U(t), describes the evolution of a quantum state over time. In your example, the time evolution operator is represented as exp(-i\hat{H}(t2-t1)).

The probability amplitude, as you correctly stated, is given by <\Psi_{2}|U(t)|\Psi_{1}>. This represents the amplitude of the quantum state \Psi_{1} evolving into \Psi_{2} over the time interval t1 to t2.

Now, for the operation exp(\hat{H})\Psi_{1}, we can use the time evolution operator to evaluate it. We can write it as exp(-i\hat{H}t)\Psi_{1}, where t is the time interval. This represents the state \Psi_{1} evolving under the Hamiltonian \hat{H} for a time t.

In summary, the time evolution operator plays a crucial role in calculating probability amplitudes and understanding the evolution of quantum states over time. I hope this helps clarify your understanding of these concepts. Please let me know if you have any further questions.