Finding the expectation value of energy using wavefunc. and eigenstate

In summary, the conversation is about a homework problem that involves using Latex and images to solve parts (a), (b), and (c). The person asking for help is unsure if they did part (a) correctly and is looking for tips to finish the last part. Another person responds with their solution for part (a) and gives a hint for part (c). The first person then responds with confusion about their answer for part (a) and asks for help with part (c). Finally, the first person figures out the solution for part (c) and thanks the second person for their help.
  • #1
Dixanadu
254
2

Homework Statement


Hey guys!

So this is a bit of a long question, I've done most of it but I need a few tips to finish the last part, and I'm not sure if I've done the first one correctly. I'll be typing it up in Word cos Latex is long!

http://imageshack.com/a/img5/8335/n7iw.jpg


Homework Equations



http://imageshack.com/a/img820/2584/viiw.jpg

The Attempt at a Solution


http://imageshack.com/a/img534/1410/lbv9.jpg

Please let me know if I've done part (a) right and a bit of a hint with part C. Thanks guys!
 
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  • #2
Hello.

Your work looks pretty good. For part (a) I get a result that differs from yours by a factor of 2.

For part (b), the high probability for ##P_1## should not be too surprising if you graph the state ##\Psi(x,0)## and compare it to the graph of ##\psi_1(x)##.

For (c), there is an expression in the "stuff we need" that you can use.
 
  • #3
Okay, I don't know why your answer differs by a factor of 2 for part (a). Unless I've made a serious mistake I don't see how.

Second, for part (c), I have no idea what to do. If I use the expression of <E> in terms of the sum of probabilites in the "stuff we need" section, I still have a factor of E_n on the RHS, don't know how to get rid of it...
 
  • #4
Oh don't worry about it TSny! I managed to figure it out. You were right, as always :D thanks for the help! I got the same answer for part (c) as you did for part (a).
 
  • #5


I appreciate your effort in solving this problem. Your approach to solving part (a) seems to be correct. However, it is always a good practice to double check your work and make sure all the calculations are accurate. As for part (c), it seems like you are struggling with it. My suggestion would be to review the equations and concepts related to expectation values and eigenstates, and try to apply them to the problem. You can also consult your textbook or seek help from your instructor or peers if needed. Keep up the good work!
 

1. What is the expectation value of energy in quantum mechanics?

The expectation value of energy in quantum mechanics is a measure of the average energy of a system in a given state. It is calculated by taking the inner product of the wavefunction of the system and the operator representing the energy of the system.

2. How is the expectation value of energy calculated using wavefunctions and eigenstates?

The expectation value of energy can be calculated by taking the integral of the wavefunction multiplied by the energy operator, with respect to the variable representing the energy. This integral is then divided by the integral of the square of the wavefunction. The resulting value is the expectation value of energy.

3. What is the relationship between the expectation value of energy and the probability of measuring a certain energy?

The expectation value of energy is equal to the average energy of a system, and it is also the most probable value of energy that would be measured if the system is in that state. However, it is possible for the measured energy to be different from the expectation value due to the probabilistic nature of quantum mechanics.

4. Can the expectation value of energy be negative?

Yes, the expectation value of energy can be negative. This indicates that the system has a higher probability of being in a lower energy state rather than a higher energy state. It is important to note that the energy operator itself can have both positive and negative values, depending on the system and the chosen reference point.

5. How does the expectation value of energy change with the change in the wavefunction or the eigenstate?

The expectation value of energy is dependent on the specific state of the system, as represented by the wavefunction or eigenstate. If the wavefunction or eigenstate changes, the expectation value of energy will also change accordingly. It is a dynamic quantity that can change as the system evolves over time.

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