Time evolution operator

In summary, the conversation discusses the use of time evolution operator, U, and its relation to the Hamiltonian represented by a 2x2 matrix, H. It is shown that U is equal to the exponential of the matrix with a missing unit matrix, and that this is true due to the power series representation of the exponential function.
  • #1
Gregg
459
0
If you have some Hamiltonian represented by a 2x2 matrix

## H = \left(
\begin{array}{cc}
0 & \Delta \\
\Delta & 0
\end{array}
\right) ##

And you want to use the time evolution operator

## U = \exp ( - \frac{i}{\hbar} H t ) ##

it says that

## U = \exp (- \frac{i \Delta}{\hbar} t) ##

Why is this?
How did the ##\Delta## get out?
 
Last edited:
Physics news on Phys.org
  • #2
The was an error in the latex.

Is it just true that

## e^A | \psi_\alpha > e^{\alpha} | \psi_{\alpha} > ##

Where

## A | \psi_\alpha>= \alpha |\psi_{\alpha}> ##?
 
  • #3
Gregg said:
The was an error in the latex.

Is it just true that

## e^A | \psi_\alpha > e^{\alpha} | \psi_{\alpha} > ##

Where

## A | \psi_\alpha>= \alpha |\psi_{\alpha}> ##?

The exponential of a matrix is really just short hand for the power series representing the exponential function, [tex]e^A = \sum_{n=0}^\infty \frac{A^n}{n!}.[/tex]
So if you apply this power series operator to the wave function, you should see that in fact yes, what you've said is true.
 
  • #4
Just a few LaTex hints: use \left( and \right) before brackets containing fractions and \langle and \rangle for bra-ket notation. (No bold in the LaTex code, of course)

As for
U=exp(−iΔt/ℏ)

in your initial post, it's incomplete, it misses the unit matrix 2x2 in the RHS.
 
  • #5
Steely Dan said:
The exponential of a matrix is really just short hand for the power series representing the exponential function, [tex]e^A = \sum_{n=0}^\infty \frac{A^n}{n!}.[/tex]
So if you apply this power series operator to the wave function, you should see that in fact yes, what you've said is true.

Thanks that's so obvious now
 

1. What is the time evolution operator?

The time evolution operator is a mathematical tool used in quantum mechanics to describe how a quantum system changes over time. It is represented by the symbol U(t), where t represents time, and is used to calculate the state of a system at a future time based on its current state.

2. How is the time evolution operator related to the Schrödinger equation?

The time evolution operator is intimately related to the Schrödinger equation, which describes how the state of a quantum system changes over time. The Schrödinger equation is solved by the time evolution operator, which transforms the initial state of the system into its state at a later time.

3. Can the time evolution operator be applied to classical systems?

No, the time evolution operator is a concept specific to quantum mechanics and cannot be applied to classical systems. Classical systems follow deterministic laws and do not exhibit the probabilistic behavior that quantum systems do, which is described by the time evolution operator.

4. How is the time evolution operator related to observables?

The time evolution operator is used to calculate the expectation values of observables in a quantum system. These expectation values represent the average measurement of a physical quantity in the system and are calculated by taking the inner product of the time-evolved state with the operator representing the observable.

5. Can the time evolution operator be used to predict the exact state of a quantum system?

No, the time evolution operator can only be used to calculate the probability of a system being in a certain state at a given time. Due to the probabilistic nature of quantum mechanics, it is impossible to predict the exact state of a system at a future time. The time evolution operator can only provide a range of possible states and their associated probabilities.

Similar threads

  • Advanced Physics Homework Help
Replies
1
Views
902
  • Advanced Physics Homework Help
Replies
9
Views
809
  • Advanced Physics Homework Help
Replies
3
Views
254
  • Advanced Physics Homework Help
Replies
10
Views
396
  • Advanced Physics Homework Help
Replies
9
Views
873
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
745
  • Advanced Physics Homework Help
Replies
0
Views
163
Replies
2
Views
960
  • Advanced Physics Homework Help
Replies
1
Views
170
Back
Top