Difference between the superposition of classical complex variable and that of a quantum variable

Question

I was wondering what is the key difference between the superposition of classical complex variables (say for ex. Electric fields) and quantum variables (say for ex. a wavefunction). Take for example the complex representation of electric field $E=E_0.e^{i\omega t}$.

Hecht says, "after expressing a wave as a complex function and then performing operations with/on that function, the real part can be recovered only if those operations are restricted to addition, subtraction, multiplication or division by a real quantity, and differentiation or integration with respect to a real variable " i.e,

$E_1+E_2 = E_{01}.e^{i\omega t}+E_{02}.e^{i\omega t}$, for which the physically meaningful total field will be calculated as $Re(E_1+E_2)$.

On the other hand if we consider a Quantum wavefunction, which is also a complex quantity, $|\psi \rangle = a+ib $. The physically meaningful quantity is not the $Re(a+ib)$, but rather the square of the modulus of the complex variable : $ |\langle \psi |\psi \rangle|^2=a^2+b^2 $

Now I think this makes all the difference in the world and adds to the so called weiredness of the quantum nature. My questions are, am I right in pointing out that this is the key difference in how we interpret 'superposition' in the two frameworks and if yes what could be the reason for the differing interpretations.

Answer 1

On the other hand if we consider a Quantum wavefunction, which is also a complex quantity, $|\psi \rangle = a+ib $.

Your equation has a "ket" on one side, and a single complex number on the other side. It does not make sense (at least not in the usual/common interpretation of the symbols you are using). A ket like $|\psi\rangle$ is a vector in a Hilbert space, which can not just be set equal to a complex number $a+ib$.

You need to include some basis vectors of your Hilbert space. I will assume you are trying to ask about a two-dimension Hilbert space (e.g., the space of a single qubit) and I will re-write your ket $|\psi\rangle$ in terms of two orthonormal basis vectors $|0\rangle$ and $|1\rangle$ of the Hilbert space as $$ |\psi\rangle = z|0\rangle + w|1\rangle\;,\tag{1} $$ where z and w are both complex numbers.

The physically meaningful quantity is not the $Re(a+ib)$, but rather the square of the modulus of the complex variable : $ |\langle \psi |\psi \rangle|^2=a^2+b^2 $

Your above equation "$|\langle \psi |\psi \rangle|^2=a^2+b^2$" is just nonsense under any reasonable interpretation. For example, if we assume you are literally trying to write $|\psi\rangle$ to denote a single complex number, then presumably $\langle \psi|$ is supposed to be the complex conjugate of $|\psi\rangle$. But, if this is the case then you are effectively squaring twice and you should have written your RHS as $(a^2+b^2)^2$. Maybe you have a typo? It is hard to tell given the senselessness of the first equation you proposed.

Using my Eq. (1), it is straight forward to show that $$ \langle \psi |\psi \rangle=|z|^2+|w|^2=1\;, $$ where, in the final equality, I have enforced the usual normalization condition. The "physical meaning" of this normalization is that if you sum up all probabilities you get 1.

There is more physical meaning contained in Eq. (1) than just the boring statement that probabilities add up to 1. The more interesting meaning is that $|z|^2$ is the probability to measure "0" and $|w|^2$ is the probability to measure "1."

Now I think this makes all the difference in the world and adds to the so called weiredness of the quantum nature.

Whatever interpretation of "weiredness [sic]" you are making is seemingly based on an equation that already doesn't make sense (your senseless equation $|\psi\rangle = a + ib$). So, you can't really draw any meaningful conclusions, weird or not.

My questions are, am I right in pointing out that this is the key difference in how we interpret 'superposition' in the two frameworks

No.

Answer 2

$\newcommand\bra[2][]{#1\langle {#2} #1\rvert} \newcommand\ket[2][]{#1\lvert {#2} #1\rangle}$ I would say that this is not the key difference in how we interpret superposition. The interpretation about what is a superposition is basically the same in electromagnetism and quantum theory. Take, for example,two electric fields linearly polarized:

$$ \mathbf{E_1} = e^{i(kx-\omega t)}\mathbf{\hat j}, \quad \mathbf{E_2} = e^{i(kx-\omega t - \pi/2)}\mathbf{\hat k}, $$

the superposition $\mathbf{E} = \mathbf{E_1}+\mathbf{E_2}$ is a circularly polarized electric field, and the interpretation is that this is a new field, with new properties. No one would never say that this resultant field is simultaneously linearly polarized in two different directions at the same time.

Now, consider two quantum states of a two level system (qubit)

$$ \ket{0}, \quad \ket{1} $$

The superposition $\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$ is again a new state, with new properties, as in the electric case. The only difference is that, when people tried to figure out what a superposition between two macroscopically distinct states [1] would be, they found nonsense. The allegory of the Schrodinger's cat [2] is the demonstration of such frustrated attempt to apply the concept of superposition for macroscopic systems. To avoid any conflict, you could be pragmatic and, as in the electric case, consider it as I said in the beginning of this paragraph: the superposition is a new state, with new properties. It does not necessarily shares similar properties with the states composing it: There are quantum states that could be obtained as a superposition of infinitely many different quantum states, as the singlet state of two qubits:

$$ \ket{\Psi^-} = \frac{1}{\sqrt{2}}\left(\ket{01} - \ket{10}\right) = \frac{1}{\sqrt{2}}\left(\ket{+-} - \ket{-+}\right) = ... $$

---

The key difference between the complex electric field and the wave function relies in the foundations of both. Electric fields are described by Maxwell equations, which are real equations. Thinking that way, there is not complex numbers in electromagnetism at all, everything could be done using real numbers, and the complex numbers are there just for convenience. But in quantum theory, the evolution of the wave function is governed by Schrodinger equation, which is a complex equation. It would mean that we should search for complex solutions.

For some time, some people believed that a real quantum theory could exist. But recent results[3] showed that it is not possible.

---

[1] With macroscopically distinct states, I mean states that could be distinguished macroscopically, like a pointer in a measurement apparatus. In the Schrodinger's cat allegory, the state of life of the cat is macroscopically distinct, you just need to open the box and look at it.

[2] I will let the wikipedia page about the Schrodinger's cat, but also there is this question and asnwers therein

[3]Quantum theory based on real numbers can be experimentally falsified, Marc-Olivier Renou et. al.

Answer 3

what is the key difference between the superposition of classical complex variables (say for ex. Electric fields) and quantum variables (say for ex. a wavefunction)

No difference, as long as superimposed functions are linear and homogeneous. Just in quantum mechanics superposition acts over wavefunctions, like :

$$ |\psi _{i}\rangle =\sum _{j}{C_{j}}|\phi _{j}\rangle $$

However, what you seem to talk about is interpretation of observables in classical and quantum physics. In electromagnetism, you could in principle also calculate $(E_1+E_2)(E_1+E_2)^*$. This will give you $(E_{01}+E_{02})^2$, i.e. sum of electric field amplitudes squared, but this really does not carry any additional information and so is redundant operation.

While in QM, (in simple cases) wavefunction $\psi$ is probability amplitude, can be negative AND does not carry any physical information, so to be able to extract any useful information - we need calculating probability density $\Psi\Psi^*=\rho(x)$ to be able to measure particle aviailability along $x$ and to compare that with calculated probability density values or with probability at finding particle between range $x \in [a,b]$ itself which is $P_{a\leq x\leq b}(t)=\int _{a}^{b}\,|\Psi (x,t)|^{2}dx$.

Answer 4

What do we measure?
Let me first point out some caveats related to the argument made in the Q.: we do believe that $\mathbf{E}$ and $\mathbf{B}$ are real fields/observables, unlike the wave function, which is a mathematical technique used for calculating observables (and which could be replaced by a different mathematical technique.) However, most of the time we do not really measure the components of the electromagnetic field, but its intensity $I\propto |\mathbf{E}|^2, |\mathbf{B}|^2$ (thinnk, e.g., about describing absorption in a photodetector by Fermi golden rule - here we already have a square of the matrix element, proportional to the field.) In this sense, electromagnetic field and wave function behave in a rather similar way... just like other wave phenomena.

When does it become "quantum"?
What makes this really quantum is what we describe as waves - electromagnetic field is already a wave classically, but electrons are not - the become quantum when, instead of Newton equations we write Schrödinger equation (i.e., and equation for waves.) In fact, in many problems Maxwell equations and Schrödinger equations reduce to Helmholtz or Laplace equations, with identical solutions - e.g., when solving for a two-slit experiment.

A step further is giving waves particle properties - e.g., by making them have a discrete energy spectrum, which is mathematically often achieved for EM field via a procedure called second quantization - which is really first quantization for the EM field, but "second" when applied to wave field describing an electron (that is a field that is already quantum.)

See also:
Difference between classical fields and wavefunctions? Relation between QM and Classical field theory
How does quantization arise in quantum mechanics?
How can blackbody radiation be explained by quantization?

Answer 5

The questions asks

am I right in pointing out that this is the key difference in how we interpret 'superposition' in the two frameworks

This is not a precise question, as it may be a matter of opinion on what difference is "the key difference". Nevertheless I will give an answer, which is: no, this is not the key difference. The reason I say this is that in quantum mechanics the wavefunction is, by axiom in the structure of the theory, a complex-number-valued quantity, where in classical electromagnetism the electric (and magnetic) field is a real-number-valued quantity. So when we use complex numbers in classical electromagnetism we are employing them as a mathematical tool which can help to express succinctly reasoning about real-number-valued quantities, whereas when we use complex numbers in quantum physics we are dealing directly with the quantities which are involved in the theory.

In classical electromagnetism things like $E^2$ have physical relevance (this is how we obtain the energy density, for example, and the intensity of an electromagnetic wave). One has to be careful because if $$ {\bf E}_1 = Re[ {\cal E}_1 ] $$ then it is usually the case that $$ {\bf E}_1^2 \ne Re[ {\cal E}_1^2 ], \qquad \mbox{and}\qquad {\bf E}_1^2 \ne |{\cal E}_1|^2 $$ whereas in quantum physics the quantity $|\psi|^2$ is, as the question says, itself the probability density. It follows that when we want to find out about energy density and Poynting vector in classical electromagnetism, the complex notation sometimes does not simplify the equations in a useful way, whereas in quantum theory one is working with complex-valued quantities anyway so complex notation is entirely suitable.

I would say the lesson is: be aware of both the uses and the limits of complex notation when working out the behaviour of real-valued quantities.