Infinite square well, Probability of measurement of particle's energy

[machofan]

Homework Statement

Homework Equations

The Attempt at a Solution

I have managed to do the first 3 parts of the questions. The last two 4 markers are the ones I am having difficulties with. I have tried using the expansion postulate which states the wavefunction is equal to the sum of the expansion coefficent "a" and the eigenfunction. Then by squaring the expansion coefficient, this should provide the probability.

The wavefunction from part 3) was found to be 2Acos(kx), and I've tried integrating this by squaring it, but I notice that's not the right way to go about this problem.

Any help would be much appreciated.

 

[tman12321]

As you said, the wavefunction Φ may be expanded in terms of a linear combination of the ψ(n). The {ψ} form an orthonormal basis. What happens if you take <ψ(1)|Φ>?

 

[machofan]

As you said, the wavefunction Φ may be expanded in terms of a linear combination of the ψ(n). The {ψ} form an orthonormal basis. What happens if you take <ψ(1)|Φ>?

If you take ψ and Φ, then integrate from a to -a, would that provide the expansion coefficient a$_{n}$? In which case, by squaring this coefficient, this would provide the probability?

 

[tman12321]

This is a basic fact in quantum mechanics. You should consult your textbook so that you understand this, because you are bound to see it over and over again.

 

[paranormal]

Hello,

The probability of measuring a particle's energy in the infinite square well potential is given by the square of the expansion coefficient for that particular energy level. In other words, the probability is equal to the absolute value squared of the coefficient "a" in the expansion of the wavefunction.

In the case of an infinite square well potential, the wavefunction is given by a sine or cosine function, depending on the specific boundary conditions. To find the probability of measuring a specific energy level, you would need to square the coefficient of the corresponding eigenfunction (sin or cos) in the expansion of the wavefunction.

For example, if the wavefunction is given by 2Acos(kx), the probability of measuring the energy level corresponding to this eigenfunction would be (2A)^2 = 4A^2.

I hope this helps. Let me know if you have any further questions.