# Apparent paradox concerning Heisenberg's uncertainty principle

I have just begun my Introduction to Quantum Mechanics course in my undergrad and I am trying to understand the uncertainty principle on a fundamental level. I think the best way to understand the fundamentals of this principle is to question the principle and understand why doing so is wrong.

Consider the following thought experiment:

I have a setup consisting of two independent regions. Region A with an electric field along the positive x-axis, lets say, and Region B with a magnetic field, also along the positive x-axis. In Region A I have a cathode and the electric field causes a potential difference thereby accelerating the electrons from rest to a particular velocity (along positive x-axis).

Let us just focus on a single electron. I know the exact velocity (magnitude and direction) to which this electron is accelerated to (energy conservation). Now this electron is going to enter the magnetic field in a direction that is parallel to the field. I know that the electron is going to move in a straight line in the magnetic field, without losing any energy as the magnetic field does zero work. So I know the exact momentum of the electron at all times as this momentum is not going to change as long as the electron is inside the magnetic field. So uncertainty in momentum is zero. Since I know the exact trajectory of the electron, the uncertainty in position is also zero and I know both of these simultaneously.

So it appears like this though experiment violates the Heisenberg uncertainty principle.

What is the flaw in my argument? Why can't this happen?

• The flaw in your argument is that electrons don't "go" in straight lines. That's an entirely classical picture of an electron and real electrons simply don't behave that way. – CuriousOne Jan 26 '16 at 6:56
• The whole story is full of many other wrong statements. For example, electrons aren't moving along straight lines in magnetic fields even in a classical theory. They move along circles or spirals, right? Also, all the claims that "the uncertainty is zero" are absolutely unjustified. To say that the uncertainty is zero means to be able to measure it, and know that the measurement is precise etc. In reality, it never is quite precise. – Luboš Motl Jan 26 '16 at 7:14
• @LubošMotl, In the classical theory, when an electron enters a magnetic field in a direction parallel to the field, its velocity and the magnetic field will make an angle of zero with each other. So cross product of those two times the charge of the electron is zero. So the magnetic force is zero. So the electron will indeed go in a straight line. – Rohit C Jan 26 '16 at 14:18
• Dear Rohit, don't forget that the electron also has a magnetic moment - it's a small magnet oriented along the direction of its spin which generally differs from the direction of the magnetic field. – Luboš Motl Jan 26 '16 at 15:22

but this isn't true. You know the electron energy has increased by $E$ eV, where $E$ is the potential difference you're using but you don't know what its energy was initially i.e. when it left the anode and before being accelerated by your field. The only way you can know the initial momentum of the electron precisely is if it's completely delocalised i.e. you don't know where it is or when it was emitted.