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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123 views

Rectangular window $\psi$ wave-function and the calculus of $\langle p^2\rangle$ for it

I'm currently considering a rectangular window $\psi$ function: $$ \psi(x) = \begin{cases}\left(2a\right)^{-1/2}&\text{for } |x|<a \\ 0&\text{otherwise.} \end{cases} $$ I am interested in ...
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1answer
216 views

Orthogonality of summed wave functions

Problem. I know that the two wave functions $\Psi_1$ and $\Psi_2$ are all normalized and orthogonal. I now want to prove that this implies that $\Psi_3=\Psi_1+\Psi_2$ is orthogonal to ...
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1answer
207 views

Virial theorem and variational method: an exercise (re-edited)

I have a hydrogen atom, knowing that its Hamiltonian has been modified turning the standard potential $$ V_{0}(r) = -\frac{Z}{r} $$ into $$ V(r) = -\frac{g}{r^{\frac{3}{2}}} $$ with $g$ a positive ...
4
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1answer
276 views

Why must $\Psi (x,t)$ go to zero faster than $\frac{1}{\sqrt{|x|}}$?

Why must $\Psi (x,t)$ go to zero faster than $\frac{1}{\sqrt{|x|}}$ as $|x|$ goes to $\infty$? According to Griffiths' Introduction to Quantum Mechanics, it must. I don't understand why, and this is ...
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1answer
315 views

Projection of states after measurement

Continuing from the my previous 2-state system problem, I am told that the observable corresponding to the linear operator $\hat{L}$ is measured and we get the +1 state. Then it asks for the ...
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1answer
150 views

Is the expression $S=K \log(\Psi)$ appearing in Schrödinger's first paper well defined?

I am currently reading Schrödinger's papers and happen to have some questions that maybe some expert in the field could clarify for me. Like what happens with $$S = K \log(\Psi)$$ when $\Psi<0$. ...
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1answer
88 views

Where does either Bohr or Heisenberg mention the idea of the wave function collapsing?

Could someone reference a paragraph written either by Heisenberg or Bohr where they mention the idea of the wave function collapsing?
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530 views

Understanding the Wave Function and Excited States

A wave function is an infinite dimensional vector space, how can it "live" in $\mathbb{R}^3$? Given the equation that is built like: $$\Psi (x,t) = \sum ^{\infty} _{n=1} c_n \psi _n (x) e^{-i E_n t / ...
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1answer
256 views

Young's double slit

Am I right to think the (general) probability distribution of photon in a double slit experiment at the screen has the form $|\psi|^2 = c e^{\alpha x^2}\cos^2(\beta x)$? (Due to the superposition of ...
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0answers
119 views

Bound states in 1D & 2D [duplicate]

Why does Mother Nature allow bound states in arbitrarily weak attractive potential in 2D but not in 3D? See, for example, this article, arXiv:math-ph/0208011.
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0answers
2k views

Solution for the Finite 2D Potential Well - Rotational Symmetry [closed]

I was searching for the eigensolutions of the two-dimensional Schrödinger equation $$\mathrm{i}\hbar \partial_t \mid \psi \rangle = \frac{\mathbf{p}^2}{2m_e}\mid \psi \rangle + V \mid \psi ...
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4answers
835 views

Is the wave function of a particle re-created after a measurement stops?

Yeah, I haven't quite understood, or been told, what happens to, for example an electron and it's wavefunction, when you stop to measure it? I mean, an electron has a wave function describing it's ...
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6answers
2k views

Why do wave packets spread out over time?

Why do wave functions spread out over time? Where in the math does quantum mechanics state this? As far as I've seen, the waves are not required to spread, and what does this mean if they do?
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3answers
567 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)= ...
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3answers
1k views

When Eigenfunctions/Wavefunctions are real?

When the Hamiltonian is Hermitian(i,e. beyond the effective mass approximation), generally under which conditions the eigenfunctions/wavefunctions are real? What happens in 1D case like the finite ...
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3answers
386 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 ...
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1answer
855 views

Simple Quantum Mechanics question about the Free particle, (part1)

I am reading Introduction to Quantum Mechanics by David Griffiths and I am in Ch2 page 59. He starts out writing the time dependent Schrödinger equation and the solution for $\psi(x,t)$ for the free ...
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3answers
210 views

Constructing solutions to the time-dependent Schrödinger's equation

The following question is from David Griffiths' Introduction to Quantum Mechanics: Problem 2.13 A particle in the harmonic oscillator potential starts out in the state $$\Psi(x,0) = A[3 ...
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2answers
130 views

Time evolution of a wavepacket

I do not understand why if $H\psi = E\psi$, then the time-evolution of the wavefunction is given by $e^{-iEt/h}\psi(x)$.
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2answers
150 views

Why does the magnitude squared of the wave function give us the probability density? [duplicate]

My question doesn't go much beyond the title: Why does $$\left | \psi \left ( x,t \right ) \right |^{2}$$ give us the probability density of something appearing at a certain location? I understand ...
3
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3answers
164 views

A misunderstanding regarding infinite square well

Here is a picture of the energy states of infinite potential well. We can see That the first level have a half wavelength which fittes with a full wave of the second level. $$\frac{ \lambda _{1} ...
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3answers
414 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 ...
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2answers
751 views

Is the free electron wavefunction stable?

The wavefunction of a free electrons is variously described as a plane wave or a wave packet. I am fairly happy with the wave packet, as it is localised. But if we change to the electron's rest ...
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1answer
4k views

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} = ...
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4answers
322 views

Does the wave nature of a particle refer to the wave function?

In quantum mechanics when we talk about the wave nature of particles are we referring in fact to the wave function? Does the wave function describes the probability of finding a particle (ex: ...
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5answers
352 views

How does a Wavefunction collapse?

I have been wondering and researching... How does a wavefunction collapse into one state?More specifically, what conditions cause a wavefunction for a quantum particle to collapse? Does this have to ...
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2answers
187 views

What is the analogy of $|x\rangle$ in quantum field theory?

Let me start from path integral formulation in quantum mechanics and quantum field theory. In QM, we have $$ U(x_b,x_a;T) = \langle x_b | U(T) |x_a \rangle= \int \mathcal{D}q e^{iS} \tag{1} $$ ...
3
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1answer
163 views

How does a unique electron probability distribution correspond to one wavefunction?

I'm reading the Wikipedia article on DFT, and it says that there is a one-to-one correspondence between the ground state particle density $$n_0(\vec{r}) = N \int \text{d}^3 r_2 \int \text{d}^3 r_3 ...
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3answers
796 views

Can the expectation value of the square of momentum be negative?

I've been solving a problem in quantum mechanics, and I was deriving the standard deviation of $P$, knowing that $\langle P\rangle=0$. Because $\Delta P=\sqrt{\langle P^2 \rangle - \langle P \rangle ...
3
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1answer
129 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 ...
3
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3answers
565 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?
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1answer
203 views

Expected value inequality

Why is $\langle p^2\rangle >0$ where $p=-i\hbar{d\over dx}$, (noting the strict inequality) for all normalized wavefunctions? I would have argued that because we can't have $\psi=$constant, but ...
3
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1answer
138 views

What does the notation $\Psi_k/(\Psi_k,\Psi_k)^{1/2} $ mean?

I am currently reading the paper "Gravitation and quantum mechanics for macroscopic objects" by F. Karolyhazy (1966). In his paper, he uses certain notation that I haven't come across before (he also ...
3
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1answer
1k views

Time Reversal Operator

I know that time reversal operator is an antiunitary operator. How does it work on wavefunctions? I believe in this way: $$T \psi (k,+)=e^{i\pi S_y/\hbar} K \psi (k,+) = \psi^*(-k,-),$$ but I am not ...
3
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2answers
132 views

How do you determine the “phase” of a hydrogen eigenfunction?

I've been reading the wikipedia article on the atomic orbitals of hydrogen. They have a nice collection of diagrams, such as this one for n,l,m = 3,1,1 This is apparently showing the wavefunction, ...
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1answer
81 views

Quantum Wavefunctions Without Space

A handful of physicists have a rather peculiar definition of 'nothing' in terms of cosmology. Their claim is that the Universe, assuming it has 0 total energy, could have arisen from nothing but ...
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1answer
193 views

Why is $\omega = \sqrt{K/m}$ valid for a quantum oscillator?

I'm working in the 3rd edition of Modern Physics by Serway, Moses, and Moyer. In 6.6, it talks about a quantum oscillator. I don't fully understand how the definition of frequency works. Now, we ...
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1answer
133 views

Why do we use $\psi$ instead of a straightforward probability?

What is the advantage/purpose of using $\psi$ for wavefunctions and getting the probability with $|\psi|^2$ as opposed to just defining and using the probability function?
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3answers
105 views

At what point is the spin determined in a Stern-Gerlach Apparatus

Consider a particle with spin that travels through a Stern Gerlach box (SGB), which projects the particle’s spin onto one of the eigenstates in the $z$-direction. The SGB defines separate trajectories ...
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2answers
1k views

Wavefunction as a combination of two stationary states - how to find those states?

Lets say we have a particle in a infinite square well which has a wavefunction like this ($A$ is some constant and $d$ is the width of the well): \begin{align} A\left[ \sin \left(\frac{2 \pi ...
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2answers
1k views

Vector representation of wavefunction in quantum mechanics?

I am new to quantum mechanics, and I just studied some parts of "wave mechanics" version of quantum mechanics. But I heard that wavefunction can be represented as vector in Hilbert space. In my eye, ...
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1answer
698 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?
3
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1answer
589 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 ...
3
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2answers
671 views

Can we impose a boundary condition on the derivative of the wavefunction through the physical assumptions?

Consider the Schrödinger equation for a particle in one dimension, where we have at least one boundary in the system (say the boundary is at $x=0$ and we are solving for $x>0$). Sometimes we want ...
3
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1answer
47 views

Plane wave expansion of cylindrical functions:Summation of the Hankel functions

I understand that; in cylindrical coordinates, the basic solutions of the Helmholtz equation are of the form Hankel function of integer order times a complex exponential term ...
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4answers
205 views

Projection of wavefunction onto basis function

I am given to believe that one way that one would could represent a wavefunction is by the expansion $$\Psi(x) = \Sigma_n \Psi_n(x) = \Sigma_n f_n\phi_n(x) \tag{1}$$ where $\{\phi_n (x) \}$ is an ...
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2answers
625 views

Linear vs. quadratic dispersion relation

In wave mechanics the dispersion relation between frequency $\omega$ and wave number $k$ is linear: $$\omega_n=c k_n$$ But in quantum mechanics, based on Schrödinger's equation, one can show that we ...
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2answers
238 views

Infinitely many degeneracy of Landau level: Countable or Uncountable?

Description of Landau levels can be found in many standard textbooks of quantum mechanics and here. Two ubiquitous solutions apply either the symmetric gauge $\vec{A}=(-\frac{1}{2}By,\frac{1}{2}Bx,0)$ ...
3
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1answer
114 views

Probability of measuring momentum [closed]

Suppose we have this wavefunction: $$ \psi = A \left( cos(kx) + cos (2kx) \right) $$ I have to find the possible results of measurement of momentum and their probabilities. Attempt For a momentum ...
3
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1answer
268 views

Harmonic Oscillator potential, proof that Gaussians remain Gaussians?

I read in several papers that for a Harmonic Oscillator Hamiltonian in the time dependent Schrödinger equation a Gaussian wave packet remains Gaussian. Unfortunately I could not find any proof for ...