A complex scalar field that describes a quantum mechanical system. The square of the modulus of the wave function gives the probability of the system to be found in a particular state.

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About the complex nature of the wave function?

1. Why is the wave function complex? I've collected some layman explanations but they are incomplete and unsatisfactory. However in the book by Merzbacher in the initial few pages he provides an ...
5
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3answers
1k views

Smoothness constraint of wave function

Is there anything in the physics that enforces the wave function to be $C^2$? Are weak solutions to the Schroedinger equation physical? I am reading the beginning chapters of Griffiths and he doesn't ...
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1answer
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Schrödinger function: Separable wave function with even potential function of x

I have done the Problem 2.1 in Griffiths' quantum mechanics, and it seems not making sense to me. What if the wave function isn't symmetric at all? Then obviously the proof doesn't work. The ...
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2answers
211 views

Relation between Wave equation of light and photon wave function?

Suppose in our double slit experimental setup with the usual notations $d,D$, we have a beam of light of known frequency $(\nu)$ and wavelength $(\lambda)$ - so we can describe it as $$ξ_0 = ...
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2answers
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Speed of a particle in quantum mechanics: phase velocity vs. group velocity

Given that one usually defines two different velocities for a wave, these being the phase velocity and the group velocity, I was asking their meaning for the associated particle in quantum mechanics. ...
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4answers
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Electrons - What is Waving?

If an electron is a wave, what is waving? So many answers on the internet say "the probability that a particle will be at a particular location"... so... the electron is a physical manifestation of ...
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4answers
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Why the statement “there exist at least one bound state for negative potential” doesn't hold for 3D case?

Previously I thought this is a universal theorem, for one can prove it in the one dimensional case using variational principal. However, today I'm doing a homework considering a potential like ...
12
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3answers
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Meaning of inner product $\langle \vec{r} | \psi(t)\rangle $

I have come across the equation which comes out of the nothing in Zettili's book Quantum mechanics concepts and applications p. 167: $$\psi(\vec{r},t) ~=~ \langle \vec{r} \,|\, \psi(t) \rangle.$$ ...
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3answers
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What is the relation between position and momentum wavefunctions in quantum physics?

I have read in a couple of places that $\psi(p)$ and $\psi(q)$ are Fourier transforms of one another (e.g. Penrose). But isn't a Fourier transform simply a decomposition of a function into a sum or ...
5
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1answer
410 views

Am I missing a trick to solving a 3D potential well problem?

I was playing around with a 3-D potential $V$ such that $V_{(r)} = 0$ for $r<a$, and $V_{(r)} = V_0>0$ otherwise. By using the Schrödinger Equation, I showed that: ...
13
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3answers
521 views

Wave/particle-duality as result of taking different limits of a QFT

There is an account on dualities in quantum field theories and string theories by Polchinski from last week http://arxiv.org/abs/1412.5704 At the end of page 4, he writes the wave/particle ...
5
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5answers
729 views

Is the wave function objective or subjective?

Here is a question I am curious about. Is the wave function objective or subjective, or is such a question meaningless? Conventionally, subjectivity is as follows: if a quantity is subjective then ...
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1answer
2k views

Variational Derivation of Schrodinger Equation

In reading Weinstock's Calculus of Variations, on pages 261 - 262 he explains how Schrodinger apparently first derived the Schrodinger equation from variational principles. Unfortunately I don't ...
2
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3answers
395 views

Who is doing the normalization of wave function in the time evolution of wave function?

In the Schrodinger equation, at any given time $t$ we should jointly add another sub equation, like $$||\psi_t(x)|| = 1$$ where $\psi_t(x) = \Psi(x,t)$, and then try to solve the two equations ...
6
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1answer
705 views

Bohr-Sommerfeld quantization from the WKB approximation

How can one prove the Bohr-Sommerfeld quantization formula $$ \oint p~dq ~=~2\pi n \hbar $$ from the WKB ansatz solution $$\Psi(x)~=~e^{iS(x)/ \hbar}$$ for the Schroedinger equation? With $S$ the ...
5
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2answers
993 views

Is it guaranteed that wavefunction is well behaved everywhere?

I don't really know much about Quantum mechanics, but would like to know one simple fact. The state function $\Psi(r, t)$ whose magnitude gives the probability density of the position of the particle ...
4
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1answer
2k views

3D Quantum harmonic oscillator

For an isotropic 3D QHO in a potential $$V(x,y,z)={1\over 2}m\omega^2(x^2+y^2+z^2).$$ I can see by independence of the potential in the $x,y,z$ coordinates that the solution to the Schrodinger ...
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2answers
420 views

Can we measure “wavefunction” of quantum particles?

We know that there is uncertainty principle, so question: can we ever measure wavefunction of particles? I do not think this is possible, but I am not sure. I guess that everything is probabilistic. ...
2
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1answer
3k views

Solving the time independent Schrodinger equation: Does a complex solution make sense?

In my notes, I have the Time Independent Schrodinger equation for a free particle $$\frac{\partial^2 \psi}{\partial x^2}+\frac{p^2}{\hbar^2}\psi=0\tag1$$ The solution to this is given, in my notes, ...
2
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1answer
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Plane wave expansion in cylindrical coordinates

I am trying to solve scattering problem in 2D and got to expand the wave function in cylindrical system which comes out to be Hankel function. Can you tell me how to expand the plane wave $\exp(i ...
8
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3answers
725 views

Is there only radial motion in the Hydrogen ground state?

The ground state of the Hydrogen atom is spherically symmetric. In other words, the wave function Psi depends only on the distance r of the electron from the nucleus. As a consequence all ...
12
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4answers
472 views

Reconstruction of “wavefunction” phases from $|\psi(x)|$ and $|\tilde \psi(p)|$

Consider a "wavefunction" $\psi(x)$, which has a Fourier transform $\tilde \psi(p)$ Suppose that we know, for each $x$, $|\psi(x)|^2$, and that we know, for each $p$, $|\tilde \psi(p)|^2$. Have we ...
5
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2answers
200 views

Inexact measurement and wavefunction collapse

As is usually said, measurement of an observable $q$ leads to collapse of wavefunction to an eigenstate of the corresponding operator $\hat q$. That is, now the wavefunction in $q$ representation is ...
4
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4answers
3k views

Why is wave function so important?

I am almost sure that the wave function is the most important figures in modern physics book. On the other hand I know that wave function even do not have a physical meaning it self alone! Why is ...
8
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7answers
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Is it wrong to talk about wave functions of macroscopic bodies?

Does a real macroscopic body, like table, human or a cup permits description as a wave function? When is it possible and when not? For example in the "Statistical Physics, Part I" by Landau & ...
4
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3answers
882 views

Comparison of 1D and 3D wave functions

When discussing the Schroedinger equation in spherical coordinates, it is standard practice in QM handbooks to point out that the radial part of the 3-dimensional wave equation bears a strong analogy ...
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3answers
563 views

Wavefunction, probability and impossible events

A friend of mine asked me a question, which I considered trivial at first, but after a while gave rise to some doubts. For instance, we have a potential well in 1 dimension defined by $$ V(x)= ...
2
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3answers
455 views

What is the rationale behind representing a state function by a complex valued function in QM?

What is the rationale behind representing a state function of an electron with a complex valued function $\Psi$. If only the probabilistic argument was required then why not represent it with just a ...
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1answer
125 views

Bohr-Sommerfeld quantization for different potentials

Let's have Bohr-Sommerfeld quantization for one-dimensional case: $$ \int \limits_{a}^{b} p(x)dx ~=~ \pi \hbar (n + \nu ). $$ Here $p(x) = \sqrt{2m(E - U)}$, $a, b$ are turning points, and the area ...
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1answer
131 views

Doubt in a certain equation of a research paper [closed]

In the given paper, I am stuck at equation (7). The equation that I am trying to solve for particle outside the well is : (1/g)*(g'') + (1/(r*g))*g' - (k_o)^2 = 0 where g = Radial wave function. r = ...
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1answer
657 views

Solution of 1-D Schrodinger equation for the potential $V(x) = -\frac{1}{|x|}$

May be this question might have already been asked but i couldn't find it, so let me know if its already there. Consider a potential, $V(x) = -\frac{1}{|x|}$ and if we apply this to a one dimensional ...
8
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2answers
265 views

How to guarantee square integrable solutions to time-independent Schrödinger's equation?

Given the time-independent Schrödinger’s equation in one dimension $$H\psi = E\psi$$ what restrictions can we place on V(x) (inside the hamiltonian) and E to guarantee that the solutions won't have ...
2
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2answers
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How to know if a wave function is physically acceptable solution of a Schrödinger equation?

How does one decide whether a wave function is a physically acceptable solution of the Schrödinger equation? For example: $\tan x$ , $\sin x$, $1/x$, and so on.
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5answers
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Hydrogen radial wave function infinity at $r=0$

When trying to solve the Schrödinger equation for hydrogen, one usually splits up the wave function into two parts: $$\psi(r,\phi,\theta)= R(r)Y_{l,m}(\phi,\theta).$$ I understand that the radial ...
3
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2answers
386 views

Why does $\ell=0$ correspond to spherically symmetric solutions for the spherical harmonics?

In quantum mechanics why do states with $\ell=0$ in the Hydrogen atom correspond to spherically symmetric spherical harmonics?
3
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1answer
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What does $\Psi^*$ mean in Schrodinger's formulation of Quantum Mechanics?

I am not a physics student. In one of my courses, some fundamental concepts of Quantum mechanics were needed, so I was going through them when I stumbled upon this. It says $$\text{probability} = ...
3
votes
1answer
630 views

Transmission and reflection

What is the transmission amplitude of a wavefunction $\phi(x)=e^{ikx}(\tanh x -ik)$? I would have thought that it is $\tanh x -ik$ since this is the factor associated with the forward travelling ...
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0answers
99 views

Quick question on sketching wavefunction in well

Usually for an infinite well, the sketch for n=3 level is this: Now I think if one side of the potential barrier is higher, the particle will be more likely to spend time on the left side than ...
0
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2answers
666 views

Normalization of Momentum Eigenfunctions: the number of particles

After finding the eigenfunctions $u_p(x)=Ce^{ipx/\hbar}$ of the momentum operator just like in this UCSD lecture notes, one seeks to normalize them, so one first tries: ...
0
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1answer
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Periodic boundary condition on a Wave Function of a Particle in a Box

Until now solving the Schrodinger Equation for a particle in a box was relatively easy because the boundaries conditions imposed zero value on the wave function at the boundaries. But now I must find ...
0
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1answer
321 views

Help me to visualize this wave equation in time, to which direction it moves?

The wave is $\bar{E} = E_{0} sin(\frac{2\pi z}{\lambda} + wt) \bar{i} + E_{0} cos(\frac{2 \pi z}{\lambda}+wt) \bar{j}$ Let's simplify with $z = 1$. Now the xy-axis is defined by parametrization ...
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9answers
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“Reality” of EM waves vs. wavefunction of individual photons - why not treat the wave function as equally “Real”?

In thinking how to ask this question (somewhat) succinctly, I keep coming back to a Microwave Oven. A Microwave Oven has a grid of holes over the window specifically designed to be smaller in ...
9
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5answers
600 views

Hilbert space vs. Projective Hilbert space

Hilbert space and rays: In a very general sense, we say that quantum states of a quantum mechanical system correspond to rays in the Hilbert space $\mathcal{H}$, such that for any $c∈ℂ$ the state ...
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6answers
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Is there a direct physical interpretation for the complex wavefunction?

The Schrodinger equation in non-relativistic quantum mechanics yields the time-evolution of the so-called wavefunction corresponding to the system concerned under the action of the associated ...
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3answers
825 views

What does the Schrodinger Equation really mean?

I understand that the Schrodinger equation is actually a principle that cannot be proven. But can someone give a plausible foundation for it and give it some physical meaning/interpretation. I guess ...
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4answers
505 views

Why complex functions for explaining wave particle duality?

I have this very bad habit of going to the scratch, discarding all the developments of a theory and worldly knowledge, and ask some fundamental (mostly stupid and naive, as some may say) questions as ...
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2answers
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Particle in a 1-D box and the correspondence principle

Consider the particle in a 1-d box, we know very well the solutions of it. I'd like to see how the correspondence principle will work out in this case, if we consider position probability density ...
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3answers
504 views

Could quantum mechanics work without the Born rule?

Slightly inspired by this question about the historical origins of the Born rule, I wondered whether quantum mechanics could still work without the Born rule. I realize it's one of the most ...
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5answers
334 views

Why do we need a wave function?

Assuming our only aim is to solve double slit experiment (or other problems that can be mapped into that). Knowing that electron does some strange thing not expected of a particle, we need a function ...
7
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1answer
352 views

Virial theorem and variational method: a question

I have an hydrogenic atom, knowing that its ground-state wavefunction has the standard form $$ \psi = A e^{-\beta r} $$ with $A = \frac{\beta^3}{\pi}$, I have to find the best value for $\beta$ ...