Quantum Question (dirac notation)

[bon]

Homework Statement

Please see attached :)

Homework Equations

The Attempt at a Solution

Hmm ok so stuck on all parts really..starting with (a), i see that we are looking for the probability that it is in state E_root6 i.e. |root6>

but how do we work this out? It's not a state of well defined energy so we can't just premultiply by the bra <root6| can we?

Sorry if I am being slow, we've only just started QM :(

 

[kuruman]

According to the statistical interpretation of quantum mechanics, what is the meaning of each of the coefficients in front of each term in equation 2.1?

 

[JesseC]

If you were to measure what state particle was in, you'd find it in one of those states, not a mixture. The probability of finding it in one of the states is given by the coefficient of the state squared.

Root 6 is about 2.5. So the question is asking you what the probability is of finding the particle in a state less than 2.5. What is the probability of finding the particle in either the n=1 or n=2 state?

 

[bon]

oh i see okay thanks. So is it 0.2^2 + 0.3^2? i.e. P(finding it in state 1) + P(finding it in state 2)?

 

[bon]

If you were to measure what state particle was in, you'd find it in one of those states, not a mixture. The probability of finding it in one of the states is given by the coefficient of the state squared.

Root 6 is about 2.5. So the question is asking you what the probability is of finding the particle in a state less than 2.5. What is the probability of finding the particle in either the n=1 or n=2 state?

sorry just thought id quote you so you could respond to my q above

thanks

 

[JesseC]

oh i see okay thanks. So is it 0.2^2 + 0.3^2? i.e. P(finding it in state 1) + P(finding it in state 2)?

To the best of my physics knowledge that is correct :D So long as we're independent of time.

You can confirm that the probability of finding the particle in at least one of those states is 1... if you square and then add all the coefficients you will get 1.

 

[bon]

Thanks okay! I'm now trying to do (b),(c),(d)

(b) : To work out mean value of energy i know i just work out the sum of En P(En) over all energies right? But how do i derive this from just the knowledge that the mean value of energy is <psi|H|psi> ?

(c): So i know i just multiply every state by e^-iEn/h t Do (a) and (b) stay the same?

(d): The system is in state |4> and it will stay in this state? Is this right??

 

[bon]

hellooo?

 

[JesseC]

Sorry, I don't have time at the moment to look over the rest of the questions. Have you tried a textbook for help?

I am preparing for exams and a lab report :( hopefully someone else can help you.

 

[bon]

okkkk no worries..anyone else??

 

[kuruman]

Thanks okay! I'm now trying to do (b),(c),(d)

(b) : To work out mean value of energy i know i just work out the sum of En P(En) over all energies right? But how do i derive this from just the knowledge that the mean value of energy is <psi|H|psi> ?

What is H|ψ> equal to?

(c): So i know i just multiply every state by e^-iEn/h t Do (a) and (b) stay the same?

Can you think of a reason that they should not stay the same?

(d): The system is in state |4> and it will stay in this state? Is this right??

What is the wavefunction of the system after the measurement? If you understand part (a) and given that wavefunction, what is the probability that the system will be found in states |1>, |2> and |3>?

 

[bon]

Can you think of a reason that they should not stay the same?

No i guess they are the same then?

What is the wavefunction of the system after the measurement? If you understand part (a) and given that wavefunction, what is the probability that the system will be found in states |1>, |2> and |3>?

Well after the measurement the state of the system is surely |4>. Won't it stay there?

 

[kuruman]

No i guess they are the same then?

You guess right.

Well after the measurement the state of the system is surely |4>. Won't it stay there?

It will unless you perturb it. Note that you can write this state as 0*|1>+0*|2>+0*|3>+1*|4>.