(This is an assumption of conservation of information – the laws of physics don’t cause information to disappear or new information to pop up out of nowhere.)įrom this we can actually derive a stronger statement, which is that all inner products are conserved over time. If two states are ever orthogonal, then they are always orthogonal. Third: Time evolution preserves orthogonality. U(ε) = I + εG (where G is some other operator). Second: Time evolution is always continuous, in that an evolution forwards by an arbitrarily small time period ε will change the state by an amount proportional to ε. U(0) = I (where I is the identity operator). Now, what are some basic things we can say about the time-evolution operator?įirst: if we evolve forwards in time by a length of time equal to zero, the state will not change. In other words, take the state Ψ at time 0, apply the operator U(t) to it, and you get back the state Ψ at time t. We express the notion of U(t) as a time-evolution operator by writing Let’s just give this operator a name: U(t). Well, since we’re dealing with vectors, we can very generally suppose that there exists some operator that will take any state vector to the state vector that it evolves into after some amount of time t. Now, we’re interested in the dynamics of quantum systems. (In particular, there is zero probability of either state being observed as the other.) And if ⟨φ|Ψ⟩ = 1, then the states are indistinguishable, and |Ψ⟩ =|φ⟩. If ⟨φ|Ψ⟩ = 0, then the states φ and Ψ are called orthogonal, and are as different as can be. Inner products between vectors are expressed like ⟨φ|Ψ⟩, and represent the “closeness” between these states.
These conjugated vectors are written like ⟨φ|. Similarly, any operator A has a conjugate operator A*. By analogy with complex conjugation of numbers, you can also conjugate vectors. The notation used for a state vector is |Ψ⟩, and is read as “the state vector psi”. These vectors are all unit length, and encode all of the observable information about the system. In quantum mechanics, the state of a system is described by a vector in a complex vector space. If you want to know more QM, I highly highly recommend Leonard Susskind’s online video lectures. I’m going to assume a lot of background knowledge of quantum mechanics for the purposes of this post, so as to keep it from getting too long. I want to present it here because I like it a lot. This video contains a really short and sweet derivation of the form of the Schrodinger equation from some fundamental principles.
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