For every x x and t t, Ψ(x, t) Ψ (x, t) is a complex number. Ψ∗ Ψ ∗ is the conjugate of that number, no more, no less. The reason it seems like sometimes it's only the t t part that gets conjugated is simply that often it is the only part of the wavefunction that is complex. Let's use your examples:

Mar 26, 2025 · The complex conjugate of a function is obtaining by replacing every occurrence of \(i = \sqrt{-1}\) in that function with \(-i\). This procedure eliminates complex numbers in all predictions because the product \(\Psi^* (x,t) \, \Psi \, (x,t)\) is always a real number.

Finally, the complex conjugate of \(z\) is defined \(z^\ast = x-{\rm i}\,y\). Just as we can visualize a real number as a point lying on an infinite straight-line, we can visualize a complex number as a point lying in an infinite plane.

A quantum particle at a single instant of time is described by a wave function (r);a complex function of position r. Again in the interests of simplicity we will consider a quantum particle moving in one dimension, so that its wave function (x) depends on only a single variable, the position x. Some examples of real-valued wave functions, which ...

Oct 13, 2020 · For complex functions you want to define a complex Hilbert space, i.e., a complex vector space with a scalar product. In complex vector spaces you have to define $$\langle \psi_1|\psi_2 \rangle=\int_{\mathbb{R}} \mathrm{d} x \psi_1^*(x) \psi_2(x),$$ so that the "square" $$\langle \psi|\psi \rangle \geq 0.$$

Ψ(r, t) is complex. It can be written in the form Ψ(r, t) = A(r, t) + i B(r, t) where A and B are real functions. Complex conjugate of Ψ is defined as Ψ* = A - iB. |Ψ|2 = Ψ*Ψ = A2+B2. Therefore |Ψ|2 = Ψ*Ψ is always positive and real.

May 20, 2014 · As you said, supposing that you have a wave function ψ(x) describing your particle in space, then [itex]\int_a^b \psi^*(x)\psi(x)dx[/itex] is the probability of finding your particle between the points a and b. It follows that [itex]\psi^*(x)\psi(x)[/itex] …

The complex conjugate of a function is obtaining by replacing every occurrence ofi= −1 in that function with−i. This procedure eliminates complex numbers in all

Complex numbers are commonly used in the study of waves, such as electromagnetic (light) waves, sound waves, and water waves. A wave can be described by a function f(x, t), called a wavefunction, which specifies the value of some physical quantity at each position x and time t.

Complex conjugate of a complex number in the standard representation is taking z=x+iy and making it z* =x-iy. Now the wave function in its simplest form is a complex function that means it has values in C (the complex numbers) so Ψ(x) is a complex number.