# How changing the basis of measurement produces a state which is not in the possible system states?

Consider we have a quantum system which has two possible states (as double-slit experiment):

• the photon could be pass through the first slit
• the photon could be pass through the second slit

And we decided to represent the first state as $$|0\rangle$$ and the second state as $$|1\rangle$$. Assume that the quantum state is in a superposition state $$|\psi\rangle=\sqrt(1/2) |0\rangle + \sqrt(1/2) |1\rangle$$. So, we have a probability 50% to get $$|0\rangle$$ as outcome and also a probability 50% to get $$|1\rangle$$ as outcome.

Then we measured the state of the system in the standard basis {$$|0\rangle,|1\rangle$$} and get $$|0\rangle$$ as outcome.

My question is, how we can get $$|+\rangle$$ as outcome if we measured in basis {$$|+\rangle,|-\rangle$$}? in other words, our system has only two states $$|0\rangle$$ (first slit) and $$|1\rangle$$ (second slit) so how by changing basis we get state which is not one of the two possible states of our system {$$|0\rangle,|1\rangle$$}?

I know that I miss something important regarding concept of measurement but I am beginner and would like to clearly understand this concept.

• Well, the polarization of a photon came be described in the basis {|H>, |V>}. So, if you pass a photon through a diagonal polarizer, how is it that it doesn't come out horizontal or come out vertical? That's the same situation so the answer is the same. – Craig Gidney Dec 26 '20 at 17:31
• @Craig Gidney The polarization of a photon could be in any possible state over the 2D space which is perpendicular on the direction of the propagation, so the system has an infinite number of states, while in my example the system has only two possible states {passing through the first slit and passing through the second slit}. So being in a diagonal state is also a possible state for the system. So, the example of polarization is not the same case, so it doesn't answer my question. – Islam Abdeen Dec 26 '20 at 18:07
• You're counting superpositions as states in the polarization case but not in the position case. Polarization only has two basis states, e.g. horizontal and vertical. All other polarizations are superpositions of those two. – Craig Gidney Dec 26 '20 at 18:38
• @CraigGidney Well, how can we get one of possible superposition states after measurement? Isn't wavefunction get collapsed after measurement and we only left with one of basis vectors/states of the photon, horizontal ($|0\rangle$) or vertical ($|1\rangle$)? I know that I misunderstand something but I don't know what is it, so detailed explanation will be fully appreciated. – Islam Abdeen Dec 26 '20 at 21:07
• The superposition can collapse to a state that's not a basis vector. – Craig Gidney Dec 26 '20 at 22:11

This is a Prologue taken from David McIntyre's Quantum Mechanics textbook (the link should lead you to the pdf version of the textbook) that I thought is pretty neat and shows the counter intuitive of QM. Also, section 1.1 describes the measurement phenomena of QM really nicely. I encourage you to read through it.

It was a dark and stormy night. Erwin huddled under his covers as he had done numerous times that summer. As the wind and rain lashed at the window, he feared having to retreat to the storm cellar once again. The residents of Erwin's apartment building sought shelter whenever there were threats of tornadoes in the area. While it was safe down there, Erwin feared the ridicule he would face once again from the other school boys. In the rush to the cellar, Erwin seemed to always end up with a random pair of socks, and the other boys teased him about it mercilessly.

Not that Erwin hadn't tried hard to solve this problem. He had a very simple collection of socks - black or white, for either school or play; short or long, for either trousers or lederhosen. After the first few teasing episodes from the other boys, Erwin had sorted his socks into two separate drawers. He placed all the black socks in one drawer and all the white socks in another drawer. Erwin figured he could determine an individual sock's length in the dark of night simply by feeling it, but he had to have them presorted into white and black because the apartment generally lost power before the call to the shelter.

Unfortunately, Erwin found that this presorting of the socks by color was ineffective. Whenever he reached into the white sock drawer and chose two long socks, or two short socks, there was a 50% probability of any one sock being black or white. The results from the black short drawer were the same. The socks seemed to have "forgotten" the color that Erwin had determined previously.

Erwin also tried sorting the socks into two drawers based upon their length, without regard to color. When he chose black or white socks from these long or short drawer, the socks had also "forgotten" whether they were long or short.

After these fruitless attempts to solve this probem through experiments, Erwin decided to save himself the fashion embarrassment, and he replaced his sock collection with a set of medium length brown socks. However, he continued to ponder the mysteries of the socks throughout his childhood.

After many years of daydreaming about the mystery socks, Erwin Schrodinger proposed his theory of "Quantum Socks" and become famous. And this is the beginning of the story of the quantum socks.

Farfetched?? You bet. But Erwin's adventure with his socks is the way quantum mechanics works.

• thanks so much. I will download it. – Islam Abdeen Dec 27 '20 at 16:07