Finding transition matrix, no % probability given

[Razberryz]

Homework Statement

Consider a quantum mechanical system with three states. At each step a particular particle transitions from one state to a different state.

Empirical data show that if the particle is in State 1, then it is 7 times more likely to go to State 2 at the next step than to State 3. If it is in State 2, then it is 4 times more likely to go to State 3 at the next step than to State 1. If it is in State 3 then it is equally likely to go to State 1 or State 2 at the next step.

Let A be transition matrix for this markov chain. Find a₃₁, a₃₂, and a₃₃ (i.e., find the last row in the transition matrix)

Homework Equations

The Attempt at a Solution

\begin{bmatrix}0 & 0.2 & 0.5 \\ 0.875 & 0 & 0.5 \\ 0.125 & 0.8 & 0 \end{bmatrix}

Am I on the right track?

 

[Ray Vickson]

Homework Statement

Consider a quantum mechanical system with three states. At each step a particular particle transitions from one state to a different state.

Empirical data show that if the particle is in State 1, then it is 7 times more likely to go to State 2 at the next step than to State 3. If it is in State 2, then it is 4 times more likely to go to State 3 at the next step than to State 1. If it is in State 3 then it is equally likely to go to State 1 or State 2 at the next step.

Let A be transition matrix for this markov chain. Find a₃₁, a₃₂, and a₃₃ (i.e., find the last row in the transition matrix)

Homework Equations

The Attempt at a Solution

\begin{bmatrix}0 & 0.2 & 0.5 \\ 0.875 & 0 & 0.5 \\ 0.125 & 0.8 & 0 \end{bmatrix}

Am I on the right track?

These numbers all look wrong to me. Where did you get them? Show your work, so that we have a basis for discussion.

Note added in edit: apparently you are using the convention that the columns sum to 1. I have only seen this convention used in about 0.01% of the things I have ever read.

However, if that is the case your matrix is correct, at least as far as the state-transition "movements" are concerned. What is absent here is any notion of time scale. A quantum system will not just make a transition religiously every single time unit; in reality, it will remain in some state for a random amount of time, then jump to another state and remain there for another random amount of time, etc. In other words, you really should use a continuous-time Markov chain, rather than a discrete-time one. Unfortunately, in this problem you were not given any information about average sojourn times in the various states, so you cannot make a realistic model of an actual quantum system.

 

[Razberryz]

These numbers all look wrong to me. Where did you get them? Show your work, so that we have a basis for discussion.

Imagine a 1 above the 1st column, a 2 above the 2nd column, and 3 above the 3rd column
do the same for the rows (1 beside the 1st row...)

and P_ij = changing from state j to state i

Basically I put 0.875 in the 2nd row and 1st column, because the transition from state 1 to 2 is 7x more likely than the transition from 1 to 2, and 0.875% is 7x more than 0.125%. I think I'm doing this totally wrong. Thanks for the reply.

 

[Razberryz]

P would be the transition matrix, with 1,2,3 as states

 

[Ray Vickson]

P would be the transition matrix, with 1,2,3 as states

I edited my original post, to recognize that you are using the quite rare convention (columns summing to 1, instead of the usual one of rows summing to 1). That being said, read the rest of my edited post, because it makes a crucial point about relevance of the model you wrote down.

 

[Razberryz]

It's the correct answer! Thank you!