What Are the Probable Values of Energy and Momentum for a Free Particle at t=0?

[nikolafmf]

Homework Statement

At time t=0 free particle is found in state psi=const*sin(3x)*exp[i(5y+z)]. What values for energy and for momentum we can get if we measure them at t=0 and with what probability?

Homework Equations

Well, we know that eigenvalues of energy and momentum operator for free particle are (hbar*k)^2/(2m) and hbar*k respectively.

The Attempt at a Solution

So, does measured momentum equal hbar*(0, 5, 1) and energy 13(hbar)^2/m? Is probability 100%? Book hints that we can get different values for momentum, not just one I stated before. How is it possible?

 

[BruceW]

You've got the y and z momentum correct. But You've not got the correct x momentum. In the wavefunction, the bit that depends on x is: sin(3x) So you've got to think of how to express this in terms of eigenstates.

 

[nikolafmf]

With the use of Fourier transform? But if my limits of integration are infinity, I get diverging integrals. What limits should I use?

 

[BruceW]

you don't need to do anything complicated. I'm sure you must have learned about how to write a sine function as a sum of two complex exponentials... (that's the hint)

 

[paranormal]

As a scientist, it is important to understand the concept of uncertainty in measurements. In quantum mechanics, the position and momentum of a particle cannot be known with absolute certainty at the same time. This is known as the Heisenberg uncertainty principle. Therefore, when measuring the momentum of a free particle at time t=0, we cannot say for certain that it will have a momentum of hbar*k. Instead, we can only determine the probability of obtaining a certain value for momentum.

Similarly, the energy of a free particle cannot be known with absolute certainty at time t=0. The wavefunction given in the homework statement represents a superposition of different energy states, and as a result, we cannot say for certain what the energy of the particle will be at t=0. The probability of obtaining a certain energy value can be calculated using the wavefunction and the energy operator.

In summary, while the suggested values for momentum and energy may be possible, it is important to keep in mind that in quantum mechanics, we can only determine the probability of obtaining certain values for these quantities.