Why do wavefunctions have a complex part in the Schrodinger wave equation?

[sarvesh0303]

I have seen that the Schrödinger wave equation is derived from the assumption that every wavefunction is of the form
ψ(x,t)=A(cos(2πx/λ-2πηt)+isin(2πx/λ-2πηt))
where η is the frequency


I can understand the real part of the equation. However, I am not able to understand the complex part of the equation. Why should every wavefunction have a complex part?
And why should that be isin( ) and not some other function?
Does anybody have proof of this interpretation of the wavefunction?

 

[ZapperZ]

This is math, as in complex algebra.

exp(ix) can be written as cos(x) + i sin(x). See

http://hyperphysics.phy-astr.gsu.edu/hbase/cmplx.html

Zz.

 

[sarvesh0303]

I know the euler formula and that explains the second part of my questopn but I am not able to understand why there is a complex part in the wavefunction and how do you get
(psi)=exp(i(2*pi*x/(lambda)-2*pi*(nu)*t))

 

[ZapperZ]

I know the euler formula and that explains the second part of my questopn but I am not able to understand why there is a complex part in the wavefunction and how do you get
(psi)=exp(i(2*pi*x/(lambda)-2*pi*(nu)*t))

Have you tried solving the Schrödinger equation for that particular scenario? It is, after all, a "standard" 2nd order differential equation.

Zz.

 

[sarvesh0303]

I did, but the Schrödinger Wave Equation is derived from this wavefunction right? So I find it obvious from that point of view that the wavefunction will be right. Is there another way of deriving the Schrödinger wave equation?

 

[ZapperZ]

I did, but the Schrödinger Wave Equation is derived from this wavefunction right? So I find it obvious from that point of view that the wavefunction will be right. Is there another way of deriving the Schrödinger wave equation?

Er.. no. That's like wagging the dog by its tail.

The starting point is the Schrödinger equation, NOT the wavefunction. The wavefunction in many real systems are seldom derivable, at least in closed form. But you can usually always write the Schrödinger equation/Hamiltonian for the system. And note, it IS a Hamiltonian, the same way you set up the Hamiltonian/Lagrangian in classical mechanics. You don't start from the solution to those first, do you?

Zz.

 

[jtbell]

Is there another way of deriving the Schrödinger wave equation?

It depends on what you mean by "derive" and what you want to use as your starting point (your fundamental assumptions or axioms).

There is no way to derive QM rigorously from classical mechanics. You have to start with something that is different from classical mechanics. There are various ways to do this. You can take the S.E. as "given." Or you can take the complex free-particle wave function as "given". Or you can make more sophisticated assumptions about operators acting in a Hilbert space. The ultimate "proof" is in how well its predictions for things that we can measure agree with experimental results.

Schrödinger himself was inspired to his equation by making an analogy between mechanics and optics: quantum mechanics is to classical mechanics as wave optics is to geometrical optics. He used that analogy to make an "educated guess" that led to what we know as the time-independent Schrödinger equation.

 

[sarvesh0303]

Thanks for the response guys! Very helpful insights. The derivation which I viewed had this assumption being made in the very beginning and then through simple mathematics,was transformed into the time-dependent Schrödinger Wave Equation! Could one of you please redirect me to a link where a proper derivation of the Schrödinger Wave Equation can be found? Also I'm still stuck up on the complex part. So could also link a webpage where I can find a plausible and comprehensible explanation for it?

 

[K^2]

Strictly speaking, I don't think you need complex wave function to solve Schrödinger's Equation, so it's not a good argument. Particle in a box has real solution. Harmonic oscillator is real. Hydrogen atom is written with complex solutions, but if I build the new orthogonal set from |ψ_m>+|ψ_-m> and |ψ_m>-|ψ_-m> (properly normalized), I again have the real valued solution set.

The actual physical reason the wave function is complex-valued is because there is a fundamental U(1) symmetry responsible for electromagnetic interaction in the underlying field theory, for which Schrödinger's Equation is a low energy approximation. However, you only need that U(1) symmetry to properly address electromagnetic interactions. Since Schrödinger's Equation describes an arbitrary particle in an arbitrary potential, the U(1) degree of freedom is not necessary.

In terms of why we use complex values for wave function in classical QM anyways, there are several reasons. The biggest one, of course, is that it properly describes observables. While I can construct purely real solutions to any SE, they don't necessarily describe all physical states. For example, free particle. A real-valued plain wave must have nodes in probability density. But probability density for a complex-valued plain wave can be entirely uniform.

There is also matter of convenience. Even in problems where real-valued solution is entirely satisfactory, it tends to be easier to work with complex exponents than a bunch of trigonometric functions. It's the same reason we tend to use complex notation in classical electrodynamics despite the fact that electric field is purely real there. And this is where the idea for using complex-valued function came from originally. It just happened to correspond to some more fundamental physics in QM.

 

[sarvesh0303]

I guess I understood but wouldn't the form of the wavefunction in the first part of the question also give nodes

 

[jtbell]

Particle in a box has real solution. Harmonic oscillator is real.

Not if you include the time-dependent part of the wave function.

 

[Hetware]

The imaginary parts as real as the real parts, as long as you treat the real parts as imaginary.

 

[K^2]

Not if you include the time-dependent part of the wave function.

I think that's actually true for all the cases I've mentioned. Yes, I was thinking specifically of time-independent solutions, where the overall phase can be simply dropped.

 

[sarvesh0303]

Could you guys suggest a source from which I can learn as much about wavefunctions as you guys know?
:)

 

[K^2]

You basically just need to learn more quantum mechanics. Classical first, then Quantum Field Theory. Of course, you also need to understand classical mechanics pretty well for the later.

What are you using to learn QM from? If it's too easy or too difficult, maybe we can recommend something better. What exactly is your background in mathematics? How about classical mechanics? Are you comfortable with Hamilton's Principle?

 

[Hetware]

I have seen that the Schrödinger wave equation is derived from the assumption that every wavefunction is of the form
ψ(x,t)=A(cos(2πx/λ-2πηt)+isin(2πx/λ-2πηt))
where η is the frequency


I can understand the real part of the equation. However, I am not able to understand the complex part of the equation. Why should every wavefunction have a complex part?
And why should that be isin( ) and not some other function?
Does anybody have proof of this interpretation of the wavefunction?

The imaginary part is just as good as the real part, just don't mix them together. The use of complex numbers just makes symbol manipulation easier.

 

[sarvesh0303]

Actually I am a high school student and am taking the edX course (3.091x Introduction to Solid State Chemistry). In the course the lecturer derived the equation using the wavefunction that I have mentioned above. I guess I should wait till I learn Classical Mechanics( I am only familiar with the basics of it) and the Quantum Mech. Then I will be able to understand about the wavefunction better. Thanks for all the responses everyone!