Uncertainty principle if only one velocity direction is known

[Janusz Bednarski]

Homework Statement

An electron moves along the x-axis with a speed of 10^5 m/s.
a. The speed in x direction is measured with an error margin of 0.1%. What is the smallest uncertainty with which one can simultaneously measure the position in the x-axis?
b. What is the smallest uncertainty with which one can simultaneously measure the position
in y direction?

Relevant Equations

∆ x ∆ p_x ≥ h/4pi

I calculated the answer for question a to be about 10^-6 m/s (1 significant figure), but I am stuck on question b. It seems to me that it is a trick question because we don't know anything about the speed in the y-direction, and the answer can be everything from 0 to infinity. Am I right?

 

[PeroK]

Note that the $\Delta$ in the HUP (Heisenberg Uncertainty Principle) is not the experimental error (as implied in this question). It's actually the variance in theoretically infinitely precise repeated measurements.

I guess that the question comes from a quantum physics course or textbook?

 

[Janusz Bednarski]

Note that the $\Delta$ in the HUP (Heisenberg Uncertainty Principle) is not the experimental error (as implied in this question). It's actually the variance in theoretically infinitely precise repeated measurements.

I guess that the question comes from a quantum physics course or textbook?

Yes, it's from an old exam in the course I'm taking right now.

 

[Hill]

The answer to the question (a) does not look right: m/s is wrong unit for position.

 

[Janusz Bednarski]

The answer to the question (a) does not look right: m/s is wrong unit for position.

Oh haha yes I meant just m.

 

[Hill]

the answer can be everything from 0 to infinity

The question is about the smallest uncertainty.

 

[PeroK]

Homework Statement: An electron moves along the x-axis with a speed of 10^5 m/s.
a. The speed in x direction is measured with an error margin of 0.1%. What is the smallest uncertainty with which one can simultaneously measure the position in the x-axis?
b. What is the smallest uncertainty with which one can simultaneously measure the position
in y direction?
Relevant Equations: ∆ x ∆ p_x ≥ h/4pi

I calculated the answer for question a to be about 10^-7 m/s (2 significant figures), but I am stuck on question b. It seems to me that it is a trick question because we don't know anything about the speed in the y-direction, and the answer can be everything from 0 to infinity. Am I right?

In QM, a free electron may be described by a wave packet, but not by a classical trajectory. How do we know the electron is moving along the x-axis unless we have measured the y-component of its position? And, according to the HUP, if we know precisely the electron's y-position, then we don't know it's y-momentum. And, if we don't know it's y-momentum, how can we know it is moving along the x-axis?

The question is simply not quantum mechanical. The HUP itself forbids the certainties and the precise classical trajectory specified in the question!

 

[Janusz Bednarski]

The question is about the smallest uncertainty.

Oh, so you think that the answer should be 0?

 

[Hill]

How do we know the electron is moving along the x-axis

I don't read it as "electron is moving along the x-axis", but rather as "its speed along the x-axis is ..." Then we really don't know anything about the y-axis.

Oh, so you think that the answer should be 0?

Yes, I do.

 

[PeroK]

I don't read it as "electron is moving along the x-axis", ...

Homework Statement: An electron moves along the x-axis with a speed of 10^5 m/s.

 

[Hill]

Right, but it does not mean (in my English which is not my native language) that it moves only / purely along the x-axis. To me, it means that whatever its movement is, its speed along the x-axis is 10^5 m/s.
IOW, "Along the x-axis, an electron moves with the speed of 10^5 m/s."

 

[PeroK]

Oh, so you think that the answer should be 0?

The textbooks from which I learned QM (Griffiths & Sakurai) both presented the theory of QM. I know from having been a Homework Helper on here for many years, that many courses teach a hybrid theory where QM is just an addendum to classical mechanics - in terms of introducing uncertainty in measurements.

I honestly don't know what the answer should be, as I've never studied this hybrid theory.

I just wanted to say, without being too negative, that you are not actually being taught QM here! QM is a radical departure from classical physics. Heisenberg is supposed to have said "anyone who is not shocked by QM hasn't understood it". In your case, it's perhaps "anyone who isn't shocked by QM isn't being taught it properly"!

Sorry I can't be of more help.

 

[haruspex]

From what little I understand of QM, the different axes cannot be treated independently. The deltas are standard deviations, and the standard deviation of a velocity variable is a magnitude derived from a 3D probability distribution of velocity vectors.

 

[Vanadium 50]

There are at least three different interpretations of "moves along the x-axis with a speed of..."

1. The x-component of its velocity is as stated
2. The direction it is moving is defined to be the x-direction
3. The particle is constrained to move in the x-direction, like a bead on a wire.

I don't think this is intended to be a physics question - it's a "can you plug in numbers" question. This is in addition to the confusing way in which it was written.