This video contains a really short and sweet derivation of the form of the Schrodinger equation from some fundamental principles. I want to present it here because I like it a lot.
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. If you want to know more QM, I highly highly recommend Leonard Susskind’s online video lectures.
So! Very brief review of basic QM:
In quantum mechanics, the state of a system is described by a vector in a complex vector space. These vectors are all unit length, and encode all of the observable information about the system. The notation used for a state vector is |Ψ⟩, and is read as “the state vector psi”. By analogy with complex conjugation of numbers, you can also conjugate vectors. These conjugated vectors are written like ⟨φ|. Similarly, any operator A has a conjugate operator A*.
Inner products between vectors are expressed like ⟨φ|Ψ⟩, and represent the “closeness” between these states. If ⟨φ|Ψ⟩ = 0, then the states φ and Ψ are called orthogonal, and are as different as can be. (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⟩.
Now, we’re interested in the dynamics of quantum systems. How do they change in time?
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. Let’s just give this operator a name: U(t). We express the notion of U(t) as a time-evolution operator by writing
U(t)|Ψ(0)⟩ = |Ψ(t)⟩
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.
Now, what are some basic things we can say about the time-evolution operator?
First: if we evolve forwards in time by a length of time equal to zero, the state will not change. (This is basically definitional.)
I.e. U(0) = I (where I is the identity 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.
I.e. U(ε) = I + εG (where G is some other operator).
Third: Time evolution preserves orthogonality. If two states are ever orthogonal, then they are always orthogonal. (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.)
I.e. ⟨φ(0)|Ψ(0)⟩ = 0 ⇒ ⟨φ(t)|Ψ(t)⟩ = 0
From this we can actually derive a stronger statement, which is that all inner products are conserved over time. (The intuition for this is that if all our orthogonal basis vectors stay orthogonal when we evolve forward in time, then time evolution is something like a rotation, and rotations preserve all inner products.)
I.e. ⟨φ(0)|Ψ(0)⟩ = ⟨φ(t)|Ψ(t)⟩
So our starting point is:
- U(t)|Ψ(0)⟩ = |Ψ(t)⟩
- U(0) = I
- U(ε) = I + εG
- ⟨φ(0)|Ψ(0)⟩ = ⟨φ(t)|Ψ(t)⟩
From (1) and (4), we get
U(t)|Ψ(0)⟩ = |Ψ(t)⟩
U(t)|φ(0)⟩ = |φ(t)⟩
⟨φ(t)|Ψ(t)⟩ = ⟨φ(0)|U*(t) U(t)|Ψ(0)⟩
U*U = I
Operators that satisfy the identity on the final line this are called unitary – they are analogous to complex numbers of unit length.
Let’s use this identity together with (3):
U*(ε) U(ε) = I
(I + εG)(I + εG*) = I
I + ε(G + G*) + ε² G*G = I
ε(G + G*) + ε² G*G = 0
G + G* ≈ 0
In the last line, I’ve used the assumption that ε is arbitrarily small, so that we can throw out factors of ε².
Now, what does this final line tell us? Well, it says that the operator G (which dictates the change in state over an infinitesimal time) is purely imaginary. By analogy, any purely imaginary number y = ix satisfies the identity:
y + y* = ix + (ix)* = ix – ix = 0
So if G is purely imaginary, it is convenient to consider a new purely real operator H = iG. This operator is Hermitian by construction – it is equal to its complex conjugate. Substituting this operator into our infinitesimal time evolution equation, we get
U(ε) = I – iεH
Now, let’s consider the derivative of a quantum state.
d|Ψ⟩/dt = (|Ψ(t + ε)⟩ – |Ψ(t)⟩) / ε
= (U(ε) – I)|Ψ(t)⟩ / ε
= -iεH|Ψ(t)⟩ / ε
Thus we get…
d/dt|Ψ⟩ = -iH|Ψ⟩
This is the time-dependent Schrodinger equation, although we haven’t yet specified what this operator H is supposed to be. However, since we know H is Hermitian, we also know that H corresponds to some observable quantity.
It turns out that if we multiply this operator by Planck’s constant ħ, it becomes the Hamiltonian – the operator that corresponds to the observable energy. We’ll just change notation subtly by taking H to be the Hamiltonian – that is, what we would previously have called ħH. Then we get the more familiar form of the time-dependent Schrodinger equation:
iħ d/dt|Ψ⟩ = H|Ψ⟩