thosgood
A remark to 1.2.1
Dear Tim,
I also would like to make a remark on some statements in 1.2.1: '..it must have, somehow, experienced the two paths...' '..because it has travelled through both...'
I assume you are familiar with the debate around this remark. Do Oxfordians deserve an extra footnote? Griffiths calls it 'quantum folklore' - a bit provocative, I agree. (Consistent Quantum Theory, 7.4 pg. 105. - see https://quantum.phys.cmu.edu/CQT/) Generalising probability theory solves the problem, as Griffiths shows (See Griffiths Ch.5). Above:
- Throwing a dice gives a probability of 1/6 to throw a 2. This means, that when you throw the dice e.g. 10.000 times you will have a 2 in about 1666 throws.
- Every throw on the other hand produces a 1, 2, 3, 4, 5 or 6.
Both descriptions of describing the reality of throwing a dice are correct.
Isn't it the same with our quantum particle? The particle always goes through one slit (upper or lower). But because of the probabilistic character of the particle itself, one has to do the experiment (exactly the same experiment) many times, to make a true description of the particle's behaviour. At the end it appears that sometimes the particle goes through the upper slit, and sometimes through the lower slit.
There is a more technical argument also. If the upper-slit is represented by $ket{0}$ and the lower-slit by $ket{1}$ these two states are orthogonal (in a quantum analysis). If $P$ is the projection onto $ket{0}$ and $Q$ onto $ket{1}$ , then $PQ = 0$, which can be translated into $P$ AND $Q$ can never be true at the same moment in time, is always false. $P$ translates to: particle is in upper-slit, $Q$ to: particle is in lower-slit. (Griffiths, Ch.4)
For me, the problem is in the intrinsic probabilistic character of the quantum particle: I fail to have an intuitive understanding for that. And allowing complex numbers in (explaining the interference effects), makes me land on Mars - complete unknown territory.
Now, one can discuss the character of the state vector: is it only a calculation-instrument or is it describing some (part of the) physical reality of the quantum particle?
Its character differs from the classical state-vector (phase space). Although I had some thoughts about the numbers used.
- Is a real number describing a physical reality? The fraction 1/6 is (a whole divided in 6 equal pieces), but is the equivalent 1.66666....? These dots are disturbing.
- $\sqrt(2)$ has an image in reality: the hypothenusa of pythagoras' unit-triangle.
- The same is true for $\pi$ (factor of a unit-circle's circumference).
- And what about the imaginary number $i$? Yes, it's the solution of an algebraic formula $x^2 = -1$. But what's the physical representation of that? We do have a physical representation for $i$: the geometrical representation $x + iy$ in the plane. In this way, the imaginary $i$ looks like a real, existing physical entity.
Griffiths qualifies the quantum state-vector as a pre-probabilistic description of a quantum system: it can be used, to calculate the probabilities of the system to be 'seen' in specific states, the probabilities of the system, to have specific properties.
Kind regards, Jan Baltussen.
