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.
2
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1answer
56 views
In the expansion of the scattered wave function, why do these two functions have the same index?
See Griffiths Quantum Mechanics, eq. 11.21. Evidently,
$$\psi(r,\theta,\phi)=Ae^{ikz}+A\sum\limits_{l,m}^{\infty}C_{l,m}h_{l}(kr)Y_{l}^{m}(\theta,\phi).$$
But I don't see why the $l$th Hankel function ...
1
vote
4answers
134 views
Why do we consider the evolution (usually in time) of a wave function?
Why do we consider evolution of a wave function and why is the evolution parameter taken as time, in QM.
If we look at a simple wave function $\psi(x,t) = e^{kx - \omega t}$, $x$ is a point in ...
0
votes
1answer
112 views
Quantum Mechanics - Orthonormality of $d_{xz}$ and $d_{yz}$ Orbitals [closed]
Recall that all spherical harmonics $Y_{l,m}$ are orthornormal. Show that the $d_{xz}$ and $d_{yz}$ orbitals are both orthogonal to each other and normalized. In answering this question, DO NOT ...
0
votes
0answers
264 views
Expectation value of a Gaussian wave packet [closed]
How can I compute the expectation value $\langle x\rangle_t$ of a Gaussian wave packet $$\psi(x,t) = \int_{-\infty}^\infty \mathrm dp \, \hat\psi(p) \exp{\frac{-\mathrm i(px - E_p t)}{\hbar}}? $$
...
2
votes
3answers
302 views
Confused over complex representation of the wave
My quantum mechanics textbook says that the following is a representation of a wave traveling in the +$x$ direction:$$\Psi(x,t)=Ae^{i\left(kx-\omega t\right)}\tag1$$
I'm having trouble visualizing ...
1
vote
0answers
174 views
Probability and probability amplitude
What made scientists believe that we should calculate probability $P$ as the $P = \left|\psi\right|^2$ in quantum mechanics? Was it the double slit experiment? How? Is there anywhere in the ...
7
votes
2answers
227 views
Superconducting Wavefunction Phase (Feynman Lectures)
In Volume 3, Section 21-5 of the Feynman lectures (superconductivity), Feynman makes a step that I can't quite follow. To start, he writes the wavefunction of the ground state in the following form ...
3
votes
1answer
309 views
Expectation values-Wavefunction
I'm a bit puzzled about an excercise in which I have to find the expectation values for position and momentum. Normally this should be pretty easy but in this case I just don't get the point.
...
4
votes
1answer
34 views
Tip of a spreading wave-packet: asymptotics beyond all orders of a saddle point expansion
This is a technical question coming from mapping of an unrelated problem onto dynamics of a non-relativistic massive particle in 1+1 dimensions. This issue is with asymptotics dominated by a term ...
5
votes
3answers
364 views
Meaning of $\int \phi^\dagger \hat A \psi \:\mathrm dx$
While analysing a problem in quantum Mechanics, I realized that I don't fully understand the physical meanings of certain integrals. I have been interpreting:
$\int \phi^\dagger \hat A \psi ...
2
votes
3answers
220 views
Is this interpretation of $\psi=\frac{1}{\sqrt{\pi a^{3}}}e^{-r/a}$ correct?
Apologies if this is stating the obvious, but I'm a non-physicist trying to understand Griffiths' discussion of the hydrogen atom in chapter 4 of Introduction to Quantum Mechanics. The wave equation ...
1
vote
2answers
128 views
What does the quantum state of a system tell us about itself?
In quantum mechanics, quantum state refers to the state of a quantum
system. A quantum state is given as a vector in a vector space, called
the state vector. The state vector theoretically ...
5
votes
2answers
307 views
Does quantum mechanics allow faster than light (FTL) travel?
Let's suppose I initially have a particle with a nice and narrow wave function[1] (I will leave these unnormed):
$$e^{-\frac{x^2}{a}}$$
where $a$ is some small number (to make it narrow). Let's also ...
2
votes
2answers
358 views
Non-Degeneracy of Eigenvalues of Number Operator for Simple Harmonic Oscillator [duplicate]
Possible Duplicate:
Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?
I'm trying to convince myself that the eigenvalues $n$ of the number operator ...
1
vote
0answers
29 views
What methods exist for us to measure the position and momentum of atoms that make up molecules?
In reference to this paper, http://iopscience.iop.org/1355-5111/8/1/014, we are able to localize atoms using homodyne measurement. Would it be too naive to consider we can measure the position of ...
7
votes
2answers
251 views
Was uncertainty principle inferred by Fourier analysis?
I would like to know: did Heisenberg chance upon his Uncertainty Principle by performing Fourier analysis of wavepackets, after assuming that electrons can be treated as wavepackets?
1
vote
2answers
342 views
What's the physical significance of the inner product of two wave functions in quantum region?
I am a reading a book for beginners of the quantum mechanics. In one section, the author shows the inner product of two wave functions $\langle\alpha\vert\beta\rangle$. I am wondering what's the ...
3
votes
2answers
145 views
Interpretation of $e|\psi|^2$ as electron density
In solid state physics the electron density is often equated to $e|\psi|^2$. However, the Sakurai says (Chapter 2.4, Interpretation of the Wave Function, p. 101) that adopting such a view leads "to ...
2
votes
2answers
791 views
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.
...
16
votes
9answers
4k views
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 ...
1
vote
1answer
126 views
How is wavefunction probability redistributed after partial wavefunction collapse?
Suppose I set up the double-slit experiment using photons as my particle. Behind the left slit I place a beam splitter that points some of the light off in the direction of a camera (represented as ...
8
votes
2answers
254 views
Wavefunction collapse and gravity
If gravity can be thought of as both a wave (the gravitational wave, as predicted to exist by Albert Einstein and certain calculations) and a particle (the graviton), would it make sense to apply ...
4
votes
1answer
176 views
Boundary conditions from single-valuedness of spherical wavefunctions
This question is a follow-up to David Bar Moshe's answer to my earlier question on the Aharanov-Bohm effect and flux-quantization. What I forgot was that it is not the wavefunction that must be ...
2
votes
1answer
90 views
Confused over the presence of 2 expressions for $\Psi(x,t)$
I'm following Griffiths' Introduction to Quantum Mechanics, and I see that he's got 2 different expressions for $\Psi(x,t)$. One of them is ...
1
vote
1answer
419 views
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 ...
0
votes
2answers
268 views
Meaning of instantaneous probability densities in time dependent wavefunctions
For a time dependent wavefunction, are the instantaneous probability densities meaningful? (The question applies for instances or more generally short lengths of time that are not multiples of the ...
1
vote
2answers
144 views
Does quantum mechanics predict instantaneous action at a distance even without entanglement?
The suggestion that quantum mechanics implies that instantaneous action at a distance occurs is normally based on the contention that this follows from the entanglement of particles that share a ...
2
votes
2answers
179 views
Measurement and uncertainty principle in QM
The Wikipedia says on the page for the uncertainty principle:
Mathematically, the uncertainty relation between position and momentum arises because the expressions of the wave function in the two ...
1
vote
2answers
468 views
Wave function of Hydrogen Atom
Wavefunction of a Hydrogen atom is expressed in eigenfunctions as:
$$\psi(\boldsymbol r,t=0)=1/\sqrt{14}(2\psi_{100}(\boldsymbol r)-3\psi_{200}(\boldsymbol r)+\psi_{322}(\boldsymbol r) ).$$
Is ...
1
vote
1answer
318 views
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 ...
1
vote
1answer
160 views
Quantum Mechanics - The Normalization of $\psi_{3,1,1}$
Show that the hydrogen atomic wavefunction $\psi_{3,1,1}$ is normalized, and that it is orthogonal to $\psi_{3,1,−1}$.
I'm not sure if I'm supposed to consider the radial part. I can show that ...
3
votes
3answers
260 views
How to compute the expectation value $\langle x^2 \rangle$ in quantum mechanics?
$$\langle x^2 \rangle = \int_{-\infty}^\infty x^2 |\psi(x)|^2 \text d x$$
What is the meaning of $|\psi(x)|^2$? Does that just mean one has to multiply the wave function with itself?
7
votes
3answers
255 views
If superposition is possible in QM, why do we often assume systems are already in their eigenstates?
My understanding is that an arbitrary quantum-mechanical wavefunction can be written as a linear combination of eigenfunctions of some Hermitian operator, most commonly the Hamiltonian; when a ...
0
votes
4answers
405 views
What does superposition mean in quantum mechanics?
What does superposition mean in quantum mechanics?
When I say $A+B=C$ (forces). I can mean push something with force $A$ + force $B$ together, and that is same as I push it with force $C$.
But when ...
2
votes
4answers
305 views
If wave packets spread, why don't objects disappear?
If you have an electron moving in empty space, it will be represented by a wave packet. But packets can spread over time, that is, their width increases, with it's uncertainty in position increasing. ...
0
votes
0answers
168 views
Force of a particles on a Potential Barrier [closed]
A particle confined by a potential wall exerts some pressure on it. More specifically, suppose that the particle moves in this potential:
$$V(x) ~=~\left\{ \begin{array}{lcc}\text{finite ...
2
votes
1answer
1k views
Bound States in a Double Delta Function Potential [closed]
Let $V(x) = −u \delta(x) - v \delta(x − a)$ where $u, v > 0$ correspond to a potential with two $\delta$ wells. Let $v > u$. If $a$ is very large, there is certainly a bound state: the particle ...
2
votes
1answer
71 views
Does wavefunction reach its largest peak near(not in) the classical forbidden region?
As we can see in the picture in this website:
http://ctz116.ust.hk/xyli2/images/animation/quchem73.html
It's strange that the bound state wavefunction always reach its largest peak near the boundary ...
2
votes
3answers
274 views
What is the rationale behind representing a state function by a complex valued function in QM?
What is the rationale behind representing a state function of an electron with a complex valued function $\Psi$. If only the probabilistic argument was required then why not represent it with just a ...
1
vote
1answer
42 views
How to compare differences in waves?
I have a series of waves that I would like to compare to one another. The measurements are two-dimensional with time on the x-axis and an intensity measurement on the y-axis.
I'd like some way of ...
0
votes
2answers
98 views
Wave function of IQH and FQH electrons
What are the wave functions of the ground state of Integer Quantum Hall (IQH) and Fractional Quantum Hall (FQH) electrons?
3
votes
1answer
261 views
wavefunction collapse and uncertainty principle
We all know that wavefunction collapse when it is observed. Uncertainty principle states that $\sigma_x \sigma_p \geq \frac {\hbar}{2}$. When wavefunction collapse, doesn't $\sigma_x$ become $0$?, as ...
2
votes
3answers
208 views
How do I integrate $\frac{1}{\Psi}\frac{\partial \Psi}{\partial x} = Cx$
How do I integrate the following?
$$\frac{1}{\Psi}\frac{\partial \Psi}{\partial x} = Cx$$
where $C$ is a constant.
I'm supposed to get a Gaussian function out of the above by integrating but don't ...
3
votes
1answer
333 views
Finding $\psi(x,t)$ for a free particle starting from a Gaussian wave profile $\psi(x)$
Consider a free-particle with a Gaussian wavefunction,
$$\psi(x)~=~\left(\frac{a}{\pi}\right)^{1/4}e^{-\frac12a x^2},$$
find $\psi(x,t)$.
The wavefunction is already normalized, so the next thing to ...
1
vote
4answers
529 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 ...
3
votes
3answers
223 views
Can a wavefunction be solved to any arbitrary precision, given enough computer time?
I learned that the wavefunction for the hydrogen atom can be solved analytically (we did the derivation in class), but that for more complicated atoms it is "impossible" to solve and that only ...
6
votes
7answers
452 views
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 & ...
2
votes
1answer
68 views
What does $\psi_j(r_i)$ mean?
I have a mean-field Hamiltonian for N electrons. The mean-field potential felt by electron $i$ at position ${\bf r}_i$ is given by
$V^{(i)}_{int}({\bf r}_i)=\sum_{j\ne i}|\psi_j({\bf r}_i)|^2$
I ...
3
votes
1answer
324 views
Even and Odd States of a 1D finite potential well
Is it possible for a particle trapped in a 1D finite potential well to evolve from a even state to an odd state and vice-versa? Why?
0
votes
0answers
61 views
Ultrashort Optical Pulses [closed]
If an ultrashort optical pulse has a complex wavefunction with central frequency corresponding to a certain wavelength and a Gaussian envelope of RMS width of a certain time period, how can I ...
