# Linearity in Quantum Mechanics that make superposition possible

As a beginner in QM, all the video lectures that i have seen talk about superposing wave functions in order to get $\psi$. But from what i know from linear algebra, the system must be linear in order for us to do this superposition.

So, what tells us that quantum systems are linear systems? Does it come out of experimental results or from some intuitive physical explanation? If it's the first, then if we treat all quantum mechanical systems as linear, how can we find a non-linear system that might exist but has not been seen in labs yet? (I mean that in this way we exclude all possibilities that a non linear quantum system might exist). If it's the second, then can you give me that intuitive explanation?

• Jun 8 '15 at 14:16
• As a side note, I see many folks asking for intuitive explanations. This is ultimately fruitless, since your intuition may be very different from mine. For example, humans appear to have built-in genetic models for very complex things like Newtonian physics (throwing things) and language. We have no such built-in models for quantum phenomena - our intuition has to be trained by wrestling with problems to build a mental model that works. Therefore, since my intuition has been developed in a different way than yours, my ability to transfer that intuition is intrinsically very limited. Jun 8 '15 at 15:50
• When i say intuition,i mean that i want you to use the basic ideas to explain to me something rather than just giving me mathematical explanations. Jun 8 '15 at 16:20
• Your question, which lacks a proper mathematical language, is open to interpretations. For example, what do you mean exactly by "quantum systems are linear systems"? Are you referring to the equations that govern the dynamics or the mathematical structure of quantum mechanics? How do you expect an answer that doesn't involve even a pinch of maths? Jun 8 '15 at 19:10

## 2 Answers

I agree with Phoenix87 that the answer must contain a little bit of maths. Basic (nonrelativistic) quantum systems are well described by the Schrödinger equation, $H \psi = E \psi$, which is a linear (partial differential) equation: It contains the wave function $\psi$ only to the first power (no $\psi^2$ etc.). Thus, if two wave functions, say $\psi_1$ and $\psi_2$, obey the equation so will their sum or a general superposition, $\alpha \psi_1 + \beta \psi_2$ with complex $\alpha$ and $\beta$. This would not be true for a nonlinear equation.

Somewhat annoyingly (as far as terminology is concerned), there is a thing called the nonlinear Schrödinger equation, which describes classical solitons and has nothing (much) to do with quantum mechanics. From time to time, people have speculated about violations of linearity and the superposition principle, but tests have so far confirmed linearity (for a recent discussion, see e.g. this Nature Physics article).

There is not a direct link between the linearity of some physical laws and the superposition of quantum mechanics. The latter is more of a special kind of linear superposition which requires some restrictions on the coefficients.

The existence of the phenomenon of superposition of states is a characteristic of quantum mechanics. In classical mechanics such a phenomenon does not exist because every irreducible representation of the corresponding algebra of observables is one dimensional. In order to have a superposition of states, higher dimensional irreducible representations are necessary, where the state space can then be identified with the projective representation vector (Hilbert) space. When at least two independent unit vectors, say $u,v$, are then available, one can construct any superposition $$w = \alpha u + \beta v,\qquad\alpha,\beta\in\mathbb C,$$ with the condition that $|\alpha|^2+|\beta|^2 = 1$. This is necessary to ensure that $w$ defines a state, i.e. a normalised linear functional on the algebra of observables. When you look at the operation of superimposing the states generated by $u$ and $v$ into the state generated by their superposition $w$ you see that this map is not linear, for once because the projective Hilbert space (which is in a one-to-one correspondence with the accessible states) is not a linear space (think of this as a map that takes two points on a sphere and spits out another point on the sphere; this analogy is not entirely perfect in this case but close enough, modulo some identifications of points).

When there is a equation that governs the dynamics in this framework, then the superposition principle applies if this equation is linear. The meaning of superposition is a bit different in this context. While you can still superimpose states in the sense above, if $u$ and $v$ are solutions of a non-linear equation, then $w$ need not be a solution of the same non-linear equation; nonetheless it is still a valid superimposition of the states $u$ and $v$.

• At a brief read I can't see anything objectionable in this answer. I would be interested to hear why the downvoter found fault with it. Jun 9 '15 at 5:27
• I didn't downvote, but I don't quite see which question in "So, what tells us that quantum systems are linear systems? Does it come out of experimental results or from some intuitive physical explanation? If it's the first, then if we treat all quantum mechanical systems as linear, how can we find a non-linear system that might exist but has not been seen in labs yet? (I mean that in this way we exclude all possibilities that a non linear quantum system might exist). If it's the second, then can you give me that intuitive explanation?" this answers.
– JiK
Jun 9 '15 at 16:38
• The OP mentions superposition of states and then wonders about the linearity of quantum mechanical systems. It seemed to me that OP was a bit confused on the uses of the word superposition in the context of quantum mechanics. Hence my answer was an attempt to clarify the possible meanings of this notion. Jun 9 '15 at 16:42