State evolution in finite suden-broading quantum well

[Guangwei Yuan]

suppose we have a single eigen-state finite potential well like following:
V(x)= V, x<-a and x>a;
0, -a<=x<=a;
say only one eigen-state within this well S1(x)

when all of sudden this potetial well broden by size of 2
V(x)= V, x<-2a and x>2a;
0, -2a<=x<=2a;
say still only one eigen-state within this new well S1(x)

Please give a suggestion how long as t as the state from S0(x) changes to S1(x). Thanks in advance. Have a nice day.

 

[chasingwind]

Yuan, you can try to look for the answer to this problem in Zeng jinyan and Qian bochu's exercises' book on Quantum Mechanics. I think it has the same exercise. Good luck!

 

[R0man]

The state evolution in a finite sudden-broading quantum well can be described using the time-dependent Schrödinger equation. In the case of a single eigen-state finite potential well, the initial state S0(x) will evolve into the new state S1(x) as the potential well is suddenly broadened.

The time it takes for this state evolution to occur can be calculated using the time-dependent Schrödinger equation and the properties of the potential well. The time it takes for the state to change from S0(x) to S1(x) will depend on the energy difference between the two states, the width of the potential well, and the strength of the potential.

To make a suggestion for the specific time, we would need more information about the potential well, such as the depth and width of the well, as well as the energy of the initial and final states. Without this information, it is difficult to accurately estimate the time it takes for the state to evolve. However, in general, the broader the potential well and the larger the energy difference between the states, the faster the state will evolve.

In conclusion, the time it takes for the state evolution in a finite sudden-broading quantum well can be calculated using the time-dependent Schrödinger equation and the properties of the potential well. It is important to have specific information about the potential well in order to accurately estimate the time it takes for the state to change from S0(x) to S1(x).