Exam in 2 days Change of basis BRA and KET

[rwooduk]

ive been revising all holidays, unfortunately I've just realized I've been finding eigenvalues using the ensemble when i may have to change basis for the exam. looks at homework questions, workshop questions... nothing!

anyway an example problem:

rewrite

|PSI> = a|0> + b|1>

in the basis

|phi1> = a|0> + b|1>
|phi2> = b*|0> + a*|1>

my attempt

im assuming that to change basis you need to use the identity operator with the new basis, in this case

I = |phi1><phi1| + |phi2><phi2|

so I|PSI> = |PSI>

which looks a bit strange to me.

ANOTHER example

write:

|PSI> = 1/SQRT2 (|0> + |1>)

in the basis:

|3> = SQRT(1/3) |0> + SQRT(2/3) |1>
|4> = SQRT(2/3) |0> + SQRT(1/3) |1>

my attempt

I = |3><3| + |4><4|

I|PSI> = |PSI>



Please could someone confirm my method is correct before i make an *** of it in the exam. any help once again appreciated!

 

[strangerep]

I have no idea what you think you're doing in your attempted solution. And I have no idea what you meant by "using the ensemble". But maybe the following will help.

|PSI> = a|0> + b|1>

in the basis

|phi1> = a|0> + b|1>
|phi2> = b*|0> + a*|1>

Treat the last 2 equation as a pair of simultaneous equations in the unknowns |0> and |1>. I.e., express |0> in terms of |phi1> and |phi2>. Similarly for |1>. Then... (I'll leave the final step to you).

 

[rwooduk]

I have no idea what you think you're doing in your attempted solution. And I have no idea what you meant by "using the ensemble". But maybe the following will help.

Treat the last 2 equation as a pair of simultaneous equations in the unknowns |0> and |1>. I.e., express |0> in terms of |phi1> and |phi2>. Similarly for |1>. Then... (I'll leave the final step to you).

I'm not sure that way would work correctly. I am using the identity operator to change basis, which I think is the correct way. (could be wrong lol)

but thanks anyway

 

[strangerep]

I'm not sure that way would work correctly.

Well, I suggested a "low-brow" method of solution, since you seemed to having trouble with the "higher-brow" way of using a resolution of identity. So maybe you should solve the problems both ways to check that they eventually give the same answer.

HOWEVER, are you sure the new bases in both examples are as per the original question (or did you just make them up). Hint: check whether the new basis states are orthonormal. Are you sure you don't have a wrong sign somewhere?

 

[paranormal]

I can confirm that your method for changing basis is correct. In order to change basis, you need to use the identity operator with the new basis, as you have correctly done in your examples. It is important to note that the identity operator represents the transformation between the old and new basis, and by applying it to the state vector, you are essentially changing the representation of the state in the new basis.

In terms of your exam preparation, it is always a good idea to double check your work and make sure you fully understand the concepts before the exam. It is also helpful to practice with different examples to solidify your understanding. Good luck on your exam!