Quantum Mechanics – Interpretation and Maths Behind the Many-Worlds Interpretation

Question

recently I started reading the book "Something deeply hidden" by Sean Caroll. In the book he talks about the many-worlds interpretation of quantum mechanics as a more elegant way of thinking about quantum mechanics instead of the usual, as he would like to call it, textbook recipe of quantum mechanics were the wavefunction collapses when a quantum system is observed.

I found this many-worlds interpretation very interesting, but I have a bit of trouble of understandig or visualizing the mathematics behind it because he doesn't really talk about the maths in his book. My problem goes as the following: Let's take as an example a quantum particle and we as the observer want to measure/observe its location. It is known that the particle can only be in 2 location let's call them location 1 and location 2.

Let's say, for the sake of simplicity, that its wave function takes on the following form
$$\psi = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2)$$
where $\psi_1$ stands for the particle being is location 1 and $\psi_2$ stand for the particle being in location 2. When we believe the interpretation of quantum mechanics where the wave function collapses upon measurement we now know that if we measure the location of the particle it has 50% chance of being in location 1 or 2. After the measurement the wave function collapses. So let's say that we have observed the particle to be in location 1 then exactly right after the measurement the wavefunction takes the form
$$\psi = \psi_1$$

Now I understand this interpretation, but I know want to approach this same problem following the many-worlds interpretation. So this many-worls interpretation says that measurement, unlike the textbook recipe, isn't something fundamental. When the observer measures the location of the particle it is just two quantum systems interacting with each other and thus getting entangled with each other. What happens is that if we measure the particle to be in let's say location 1, then there is also anothere version of ourselves with which we don't interact that has measured the particle to be in location 2. I understand the logic behind it when it is said in words, but I want to translate it into maths. So before the measurement we still have a wave function of the form
$$\psi = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2)$$
Now with the many-world approach we can't say the wavefunction collapses upon measurement because that simply doesn't happen following this approach. So let's say in our world we observe the particle to be in location 1, then we now know that there must be another world in which the other version of ourselves observed the particle in location 2. Now how does the wavefunction of the particle look in our world? It can't be $\psi_1$ since this would implie a collapse of the wavefunction. Is it right to say it this has the above form but our world just lives inside the $\psi_1$ part of the total wavefunction?

So in short I just have problems with mathematically formulating the above example following the many-worlds approach. Any help with this would be greatly appreciated :))

P.s: I'm sorry if my way of writing isn't clear

Answer 1

One of key elements of the many worlds approach you need to consider the observer as a quantum system, with their own wavefunction. Let us say that the observer intially has a wavefuntion $\Phi_i$ and if they measure the system to be in state 1 (respectively 2) they will have a wavefunction of $\Phi_1$ (respectively $\Phi_2$). The combined system of the original system with the observer then initially $$ \Phi_i \psi = \frac{1}{\sqrt{2}}\Phi_i\left(\psi_1 + \psi_2\right)\;. $$ After measurement the observer will have become entangled with the system and we will have a state $$ \frac{1}{\sqrt{2}}\left(\Phi_1\psi_1 + \Phi_2\psi_2\right)\;. $$ That is the wavefunction has two terms ("worlds") in one of which the observer found the system in state 1 and in the other they found it in state 2. This state is entangled, so we cannot factor the state to obtain a wavefunction for the system alone.

If we wish to describe the system after measurement without direct reference to the observer we must move to a density matrix formalism. The resulting density matrix for the system, after tracing out the observer, is identical to the one obtained from a projective measurement in the standard formalism