Complex Numbers in Wave Function: QM Explained

In summary, the complex number i=√–1 is put in the wave function for matter in order to obtain wave-like behavior instead of simple dissipative dynamics. It allows for solutions that oscillate and reflect from walls. Without it, the solution would not be able to be normalized. Additionally, the time component of the wave function allows for exponential decay in various decay processes.
  • #1
Zaya Bell
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I just need to know. Why exactly what's the complex number i=√–1 put in the wave function for matter. Couldn't it have just been exp(kx–wt)?
 
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  • #2
Zaya Bell said:
I just need to know. Why exactly what's the complex number i=√–1 put in the wave function for matter. Couldn't it have just been exp(kx–wt)?

It's not possible to normalize that kind of a real-valued exponential function. The Schroedinger equation is a complex diffusion equation, and the complex numbers make it possible to obtain wave-like behavior (wavepackets reflecting from walls, etc.) instead of the simple dissipative dynamics of ordinary diffusion where the solution is a real function.
 
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  • #3
Is it a periodic function without the ##i## which we would expect from a wave? You may want to look up the connection between the sine (cosine) function and the exponential function.
 
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  • #4
Zaya Bell said:
I just need to know. Why exactly what's the complex number i=√–1 put in the wave function for matter. Couldn't it have just been exp(kx–wt)?
Sure, you can have parts of the wavefunction look like exp(kx-wt), but not the whole thing. If you consider a WKB approximation for the time independent Schrodinger equation, you get solutions that look like
##\exp(\sqrt{2m(V-E)})##
which looks like an exponential where V > E and an oscillation where E > V. You aren't allowed to have V > E for the whole wavefunction, because, as hilbert2 noted, it is not possible to normalize. But, for a bound state of a delta potential (V < 0), you have a double exponential solution https://en.wikipedia.org/wiki/Delta_potential.

As far as the time part, you get exponential decay for various decay processes.
 
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Thank you all.
 

1. What are complex numbers in wave function?

Complex numbers in wave function are mathematical constructs used in quantum mechanics to describe the state of a system. They consist of a real part and an imaginary part, and are represented by the letter "i" which stands for the square root of -1.

2. Why are complex numbers necessary in quantum mechanics?

Complex numbers are necessary in quantum mechanics because they allow us to describe the behavior of particles at the atomic and subatomic level. They help us understand the wave-like nature of particles and their probabilistic behavior, which cannot be accurately described using classical mechanics.

3. How do complex numbers relate to the wave function?

The wave function is a mathematical function that describes the state of a quantum mechanical system. It is represented by a complex-valued function, where the square of the absolute value of the function gives the probability of finding a particle in a specific state.

4. Can complex numbers be visualized in the context of wave function?

No, complex numbers cannot be visualized in the traditional sense. They do not have a physical representation but are used as mathematical tools to describe the behavior of particles at the quantum level. However, we can use visualizations such as graphs and diagrams to better understand the concepts related to complex numbers in wave function.

5. Are complex numbers used in any other areas of science?

Yes, complex numbers are used in various areas of science, such as engineering, physics, and mathematics. They have applications in fields like signal processing, electromagnetism, fluid mechanics, and more. They are also used in computer science for coding and encryption.

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