The Heisenberg uncertainty principle states that
ΔxΔp≥ℏ2
where
Δx is the uncertainty in the position,
Δp is the uncertainty in linear momentum, and
ℏ=1.054571800(13)×10−34 J s[source] is the reduced Planck constant
This means that, regardless of what quantum mechanical state the particle is in, we cannot simultaneously measure its position and momentum with perfect precision. I read that this is intrinsically linked to the fact that the position and momentum operators do not commute: [ˆx,ˆp]=iℏ.
How can I derive the uncertainty principle, as given above?
The proof I will use is taken from Griffiths, Introduction to Quantum Mechanics, 2nd ed., pp 110-111.
Defining "uncertainty"
Let's assume that the normalised state |ψ⟩ of a particle can be expanded as a linear combination of energy eigenstates |n⟩, with ˆH|n⟩=En|n⟩.
|ψ⟩=∑ncn|n⟩
The expectation value (the "mean") of a quantity, such as energy, is given by
⟨E⟩=⟨ψ|H|ψ⟩
and the variance of the energy can be defined analogously to that used in statistics, which for a continuous variable x is simply the expectation value of (x−ˉx)2:
σ2E=⟨(E−⟨E⟩)2⟩
The standard deviation is the square root of the variance, and the "uncertainty" refers to the standard deviation. It's more proper to use σ as the symbol, instead of Δ, and this is what you will see in most "proper" texts.
σE=√⟨(E−⟨E⟩)2⟩
However, it's much easier to stick to the variance in the proof. Let's generalise this now to any generic observable, A, which is necessarily represented by a hermitian operator, ˆA. The expectation value of A is merely a number, so let's use the small letter a to refer to it. With that, we have
σ2A=⟨(A−a)2⟩=⟨ψ|(ˆA−a)2|ψ⟩=⟨ψ|(ˆA−a)|(ˆA−a)ψ⟩=⟨(ˆA−a)ψ|(ˆA−a)|ψ⟩=⟨(ˆA−a)ψ|(ˆA−a)ψ⟩
where, in going from (7) to (8), I have invoked the hermiticity of (ˆA−a) (since ˆA is hermitian and a is only a constant). Likewise, for a second observable B with ⟨B⟩=b,
σ2B=⟨(ˆB−b)ψ|(ˆB−b)ψ⟩
The Cauchy-Schwarz inequality
...states that, for all vectors f and g belonging to an inner product space (suffice it to say that functions in quantum mechanics satisfy this condition),
⟨f|f⟩⟨g|g⟩≥|⟨f|g⟩|2
In general, ⟨f|g⟩ is a complex number, which is why we need to take the modulus. By the definition of the inner product,
⟨f|g⟩=⟨g|f⟩∗
For a generic complex number z=x+iy, we have
|z|2=x2+y2≥y2(since x2≥0)
But z∗=x−iy means that
y=z−z∗2i|z|2≥(z−z∗2i)2
and plugging z=⟨f|g⟩ into equation (15), we get
|⟨f|g⟩|2≥[12i(⟨f|g⟩−⟨g|f⟩)]2
Now, if we let |f⟩=|(ˆA−a)ψ⟩ and |g⟩=|(ˆB−B)ψ⟩, we can combine equations (9), (10), (11), and (16) to get:
σ2Aσ2B=⟨f|f⟩⟨g|g⟩≥|⟨f|g⟩|2≥[12i(⟨f|g⟩−⟨g|f⟩)]2
Expanding the brackets
If you've made it this far - great job - take a breather before you continue, because there's more maths coming.
We have1
⟨f|g⟩=⟨(ˆA−a)ψ|(ˆB−b)ψ⟩=⟨ˆAψ|ˆBψ⟩−⟨aψ|ˆBψ⟩−⟨ˆAψ|bψ⟩+⟨aψ|bψ⟩=⟨ψ|ˆAˆB|ψ⟩−a⟨ψ|ˆBψ⟩−b⟨ˆAψ|ψ⟩+ab⟨ψ|ψ⟩=⟨ψ|ˆAˆB|ψ⟩−ab−ab+ab=⟨ψ|ˆAˆB|ψ⟩−ab
Likewise,
⟨g|f⟩=⟨ψ|ˆBˆA|ψ⟩−ab
So, substituting (24) and (25) into (19),
σ2Aσ2B≥[12i(⟨ψ|ˆAˆB|ψ⟩−⟨ψ|ˆBˆA|ψ⟩)]2=[12i(⟨ψ|ˆAˆB−ˆBˆA|ψ⟩)]2
The commutator of two operators is defined as
[ˆA,ˆB]=ˆAˆB−ˆBˆA
So, the term in parentheses in equation (27) is simply the expectation value of the commutator, and we have reached the Robertson uncertainty relation:
σ2Aσ2B≥(12i⟨[ˆA,ˆB]⟩)2
This inequality can be applied to any pair of observables A and B.2
The Heisenberg uncertainty principle
Simply substituting in A=x and B=p gives us
σ2xσ2p≥(12i⟨[ˆx,ˆp]⟩)2
The commutator of ˆx and ˆp is famously iℏ,3 and the expectation value of iℏ is of course none other than iℏ. This completes the proof:
σ2xσ2p≥(12i⋅iℏ)2=(ℏ2)2σxσp≥ℏ2
where we have simply "removed the square" on both sides because as standard deviations, σx and σp are always positive.
Notes
1 I have skipped some stuff. Namely, ⟨ˆAψ|ˆBψ⟩=⟨ψ|ˆAˆB|ψ⟩ which is quite straightforward to prove using the hermiticity of both operators; ⟨ψ|ˆA|ψ⟩=a; ⟨ψ|ˆB|ψ⟩=b; and a=a∗ since it is the expectation value of a physical observable, which must be real.
2 This does not apply to, and cannot be used to derive, the energy-time uncertainty principle. There is no time operator in quantum mechanics, and time is not a measurable observable, it is only a parameter.
3 Technically, it is a postulate of quantum mechanics. (If I am not wrong, it derives from the Schrodinger equation, which is itself a postulate.)
What is the difference between both methods ?
