Understanding superposition of states

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I am wondering if my understanding of superposition concept is correct. Forgive me for not using QM braket notation, I am new on this site and don't know how to embed it in the post.
What confuses me about superposition concept is that people often say that some system can be in two (or more) states at the same time. Eg. electron in double slit experiment went through the both slits at the same time. Or it is said that Schrodinger's cat is in superposition of LIVE and DEAD states meaning that it is both alive and dead at the same time.

Let's say that we have a quantum system (eg. some particle) in state S = aX + bY where X and Y are eigenstates and a,b are real numbers between 0 and 1.

If we now prepare many many of these particles that are all in this superpositioned state S and we start measuring them, then each measurement will give as a result either a state X or state Y. The probability of particle being in state X corresponds to the number "a" and for particle to to be in state Y corresponds to the number "b".

So does this mean that when we say "particle is in superposition of states", we mean that "we don't know exactly what state we will get at measurement but we are sure that in some cases we will get one eigenstate and other times we will get the other eigenstate".
Is this correct "layman" thinking about the superposition? In other words, superposition just means that there is a set of possible states of the system and all we know is probably that system is one of them. IMHO it is not correct to think that one single particle is in all possible states at the same time just because we describe it as superposition of eigenstates. However, if if we talk about collection of prepared particles (rather than the single one) then we might say that. Because in that case there is a large number of prepared particles and it is very likely that all states will be given at measurement.
Please correct me if my reasoning is wrong.
 

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You are right that a superposition of A and B does mean that if we measure we might get A and we might get B. However, that superposition is a quantum state in its own right, and it has no classical equivalent so we cannot describe it using classical language. In particular, the superposition does not mean "it is either A or B but we don't know which" or "it is both A and B". Both those descriptions are common in the pop-sci press, and both are seriously misleading; you have to unlearn them before you can start learning QM for real.

It's worth noting that superposition is a property of how we choose to write the state - it's not as if some states are superpositions and some aren't. For example, you'll hear people saying something like "this particle is in a superposition of spin-up and spin-down; after we measure it the wave function will collapse to one or the other" and be tempted to think that that spin-up isn't a superposition. But it is. 100% spin-up is a 50-50 superposition of spin-left and spin-right, and the initial 50-50 up-down superposition is a 100% spin-left state. So we can choose to write both states as superpositions or not; we'll do whichever is most convenient for calculating the expected result of the measurement we're actually going to make.
 

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