Imaginary number in Schrödinger equation I am trying to understand why the wave function needs the imaginary number i. As far as I know, i was introduced to solve algebraic problems like x*x = -2 . And the wave equation uses a complex exponentional function in order to discribe the oscillatory behaviour. If someone could explain in easy terms how schrodinger found this, you could make me a happy man. Am I right to think that physicists don't like imaginary numbers? Because, by taking the absolute square of the function, the imaginary number disappears and voilà, we have a probability distribution. In fact, i'm just trying to get a better view on the Copenhagen interpretation, and it seems to me that the statistical interpretation of psi has to do with the imaginary number.
I don't think so. They show up all the time in all sorts of useful places: Fourier transforms, Laplace transforms, impedances, Lorentz invariants, etc. -Dale
Yes, they make it easier to solve differentials etc. but in fact, they can equally be solved without complex numbers, although that would make it a lot more work. I read that the schrödinger wave equation cannot be written without using an imaginary number. so that's my question
I don't actually know enough QM to address your real question. I just don't get the impression that imaginary numbers are disliked in any way. -Dale
Hi c'est moi, First, physicists do not fear or dislike complex numbers. In fact, I would say the opposite: physicists love complex numbers. They make life easy. Second, the Schrodinger equation can definitely be written without complex numbers! To do so, write the wavefunction as a sum of a real and an imaginary part. The Schrodinger equation as normally written is a complex equation, but this just means the real and imaginary parts must be separately satisfied. You obtain two coupled real differential equations relating the real and imaginary parts of the wavefunction, both of which are by definition real. Complex numbers have completely disappeared and you deal only with real functions. You can think of the real and imaginary parts as forming the components of a real two dimensional vector. The probability is then related to the magnitude of this vector, a quantity which you can readily verify reproduces the familiar formulas. A related approach is to decompose the wavefunction into an amplitude and a phase. One can again obtain purely real equations relating these two real functions. It was just such an approach, motivated by an analogy with geometric optics, that helped Schrodinger find his equation in the first place. One can also speak in terms of probability amplitudes, and complex numbers find a natural place in that formalism.
Are you refering to taking the absolute square ("seperately satisfied" )? the imaginary number will be gone then. That's how they use it, as far as I know. Yet, the imaginary number is needed because for quantum waves, kinetic energy can be negative. You can't get that with a normal description using sin and cos. Since the wavenumber (from which you get the frequency) is related to the energy of the wave, you need to get two distinguishable wave descriptions, one negative, one positive (i and -i). Then, what is the significance of the amplitude? the intensity becomes a probability, (after the absolute square is taken) and only real numbers are left, which are measureable. Thus, it seems to me that the copenhagen interpretation gives these quantum waves a contradictive nature (they're real and unreal at the same time, but more unreal), because the imaginary part is not measureable. And since the probability (intensity) deals only with real numbers, i.e. what can be measured, the real form of the wave function( WITH the imaginary parts) is just what it is, unmeasureable and thus nonsensical to talk about. I'm really just trying to understand the reasoning, it's not that I find it so important to go against this interpretation. What's more important for me is that I get a grip on what it really means. The wave picture may seem attractive at first, but if you know that every particle needs a wave description in its own 3 dimensions (i think it's called phase space), thus, 3 particles will give 3 x 3 = 9 dimensions ... it becomes clear that the intuitive attraction is just superficial. Yet, it is that same description that explains perfectly the atomic structure, and predicts quantum entanglement (not to forget interference of matter waves) ... in other words, they may not be normal waves, but there certainly is a very real aspect about them. I don't think we need to measure something in order to determine its realness. It seems like a poor criterium to me. Certainly if it has real, measureable results.
You seem to have fallen under the impression that Real Numbers are 'real' and Imaginary Numbers are some how 'imaginary'. The names are just names and have nothing to do with the 'realness' of them in the sense of any physical object. As has been mentioned, it is perfectly possible to discuss these things without resorting to using 'i' at all, but it simplifies life to have it around. (Oh, and i and -i are not positive and negative; complex numbers are not (naturally) ordered.)
I keep reading everywhere where I look into courses on QM that the schrodinger equation needs a complex plane and CAN NOT be described without resorting to it. This in contrast to other applications, where it makes the work easier. What could "i" possibly be representing in physical terms when performing a measurement? Indeed, measuring supposedly causes a collapse of a wave (neither measurements, nor collapse can be defined), the result being the absolute square root of psi. For a complex number, this is the product of the complex number itself and its complex conjugate. The "i" is gone. Heisenberg needed the "i" in his matrix QM, and a little later Schrodinger also when he developed his wave model. If I could get a minimal insight as to why they need it, that would be awesome. People can say what they want, ie that all numbers are imaginary, but that's an excuse, as apparently, these numbers work out quite well in physics. The fact that QM works, and uses "i" (and cannot do without), seems to suggest that the mathematical trickery is more than trickery but a reality we can't measure.
You're looking for some "deeper meaning" and there isn't any. i is simply a way of saying "at right angles to" . Google Euler's equation. (Euler had a lot of equations, look for one relating sines and cosines to imaginary exponents of e.) Long story short, imaginary numbers are a way to do vector operations.
I know Euler's equation. So the exponential complex wave equation can be decomposed in a cosine and a sine times i. And I know that it is represented with an Agrand diagram (imaginary axis, and real axis representing the complex plane). If you represent a wave on a normal Cartesian plane, you will see that even if you make the wavenumber k and the frequency ω negative, they will look the same as the positive form -> cos (-kx + ωt ) = cos [- ( kx - ωt )] = cos (kx-ωt) Rewrite that using i in the exponential form and you get two distinguishable states: ψ = exp [-i ( kx - ωt )] which is different from exp [i ( kx - ωt )] because now you can represent the "negative" and the "positive" form on the imaginary axis, producing two clearly distinct waves (vectors). And take the absolute square of them, and you will get the answer corresponding to 1 This signifies that the 'intensity' (in classical terms) is uniform throughout space. It can be anywhere. I suppose this is where the statistical interpretation comes from. Am I right to say that since the previous 'position' held by the amplitude had been taken by the frequency (E=hv=h-bar ω ), there was really nothing left but a statistical significance related to the observer effect. With a higher momentum, the chance for detecting a wave becomes smaller. Of course, with quantum entanglement, it appears that there is not even an observer effect involved. It's an undivided system.
