How Does the Time Evolution of Expectation Shape Scientific Understanding?

[OGrowli]

 

[planck42]

Are you asking about how quantum mechanical expectation values evolve with time? If so, then it evolves according to the differential equation

[tex]\frac{d}{dt}<{\psi}|O|{\psi}> = \frac{i}{\hbar}<{\psi}|[H,O]|{\psi}> + <{\psi}|\frac{{\partial}O}{{\partial}t}|{\psi}>[/tex]

With O being a Hermitian operator.

 

[OGrowli]

Are you asking about how quantum mechanical expectation values evolve with time? If so, then it evolves according to the differential equation

[tex]\frac{d}{dt}<{\psi}|O|{\psi}> = \frac{i}{\hbar}<{\psi}|[H,O]|{\psi}> + <{\psi}|\frac{{\partial}O}{{\partial}t}|{\psi}>[/tex]

With O being a Hermitian operator.

i don't know what happened to my original post, but I am having an issue with the following problem:

Show that:

\frac{d}{dt}<x^2> =\frac{1}{m}(< xp_x> +<p_xx>)

for a three dimensional wave packet.

relevant equations:

Ehrenfest Theorem:(1)
[tex]i\hbar\frac{d}{dt}<O>=<[O,H]>+i\hbar<\frac{\partial }{\partial t}O>[/tex]

where O is an operator
(2)
[tex]\frac{d}{dt}\int_{V}d^3r\psi ^*O\psi[/tex]

I tried using both ways illustrated above and I arrived at the same answer:

[tex]\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[\bigtriangledown ^2(x^2\psi)-x^2\bigtriangledown ^2\psi][/tex]

[tex]=\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[\frac{\partial }{\partial x}
(x^2\psi'+2x\psi)-x^2\frac{\partial^2 }{\partial x^2}\psi][/tex]

[tex]=\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[
2\psi+2x\frac{\partial }{\partial x}\psi+2x\frac{\partial }{\partial x}\psi+(x^2-x^2)\frac{\partial^2 }{\partial x^2}\psi][/tex]

[tex]=\frac{1}{m}\int_{V}d^3r[\psi ^*(-i\hbar)\psi+\psi^*x(-i\hbar\frac{\partial }{\partial x})\psi+\psi^*(-i\hbar\frac{\partial }{\partial x})x\psi][/tex]

[tex]=\frac{1}{m}(<xp_x>+<p_xx>)-\frac{i\hbar}{m}[/tex]

Am I doing anything wrong? Where does the extra term come from, and does it mean anything?

 

[OGrowli]

nvm, I got it:

[tex]
=\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[\frac{\partial }{\partial x}
(x^2\psi'+2x\psi)-x^2\frac{\partial^2 }{\partial x^2}\psi]
[/tex]

[tex]=\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[x^2\frac{\partial^2 }{\partial x^2}\psi+2x\frac{\partial }{\partial x}\psi+2\frac{\partial }{\partial x}(x\psi)-x^2\frac{\partial^2 }{\partial x^2}\psi][/tex]

[tex]=\frac{1}{m}\int_{V}d^3r[\psi^*x(-i\hbar\frac{\partial }{\partial x})\psi+\psi^*(-i\hbar\frac{\partial }{\partial x})x\psi][/tex]

[tex]=\frac{1}{m}(<xp_x>+<p_xx>)[/tex]

I went wrong thinking I could just rearrange the derivatives.

 

[paranormal]

The concept of time evolution of expectation is a fundamental aspect of many scientific fields, including physics, chemistry, and biology. It refers to the changes in the expected value of a particular quantity over time, and it plays a crucial role in understanding the behavior of systems.

In physics, the time evolution of expectation is described by the laws of quantum mechanics, which govern the behavior of particles at the atomic and subatomic levels. According to these laws, the expectation value of a physical observable, such as position or momentum, can change over time due to the inherent uncertainty of quantum systems.

In chemistry, the time evolution of expectation is essential in studying the dynamics of chemical reactions. The expected concentrations of reactants and products can change over time due to factors such as temperature, pressure, and catalysts. Understanding the time evolution of these expectations is crucial in designing and optimizing chemical processes.

In biology, the concept of time evolution of expectation is applied in studying the growth and development of living organisms. The expected traits and behaviors of an organism can change over time due to genetic and environmental factors. By understanding the time evolution of these expectations, scientists can gain insights into the mechanisms of evolution and adaptation.

Overall, the time evolution of expectation is a critical concept in science, and its study has led to significant advancements in various fields. By continuously monitoring and analyzing the changes in expectations over time, scientists can gain a deeper understanding of the fundamental laws and processes that govern our universe.