Questions tagged [wavefunction]
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. DO NOT USE THIS TAG for classical waves.
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"Forbidden zones" of a quantum particle trapped in a harmonic oscillator potential
Let's consider a confinant potential, of the form of a harmonic oscillator ($V(x) = x^2$ for instance):
Consider a classical particle which has a kinetic energy $E_i$ at $x=0$. It will forever stay ...
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Contradiction in my understanding of wavefunction in finite potential well
Most things like to occupy regions of lower potential. So the probability amplitude should be higher in a region of lower potential. I denote the potential by V.
However, we also know that the kinetic ...
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Question on computation with the bra-ket-notation
I have a rather silly question I am afraid... I am just getting to know the bra-ket-notation and still think I did not quite get it...
I want to compute a certain term, which contains the braket ...
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Question regarding gravity and quantum entanglement
We know that the Earth is revolving in an elliptical orbit around the Sun. However, if the Sun suddenly disappeared, the information would then travel to the earth, at the speed of light, and the ...
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In Quantum Mechanics is it possible to apply time evolution operator to wavefunction?
If I consider a wavefunction that is the superposition of Hamiltonian eigenfunctions, for example like: $$\psi(x)=\frac{1}{\sqrt{2}}\psi_1(x)+\frac{1}{\sqrt{2}}\psi_2(x)$$ with $\hat{H}\psi_1(x)=E_1\...
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About solutions of Schrodinger equation [closed]
If $\psi_1$ and $\psi_2$ are two independent solutions of the time independent Schrodinger equation, then is the product $\psi_1\psi_2$ also a solution of the same Schrodinger equation?
If it's not ...
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Probabilities of eigenfunctions
I am struggling to understand how to get the probabilities of each eigenstate occurring from a wavefunction that is a linear combination of eigenfunctions. If we have a wavefunction
$$\Psi = A ( e^{...
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Delta function: Intuitive way for boundary conditions
Giving the Schrödinger equation
$$-\dfrac{\hbar^2}{2\,m}\,{\partial_x}^2\psi(x)+ V(x)\,\psi(x) = E\,\psi(x)$$
with potential $V(x) = V_0\,\delta(x)$.
Solving this equation using an ordinary Ansatz ...
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Pure state vs mixed state in this example
Consider, I have a quantum state $|\Psi\rangle$, such that :
$$|\Psi\rangle=c_1|\psi_1\rangle+c_2|\psi_2\rangle$$
This is defined as a pure state, since I have complete information about the system. ...
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Why is this a negative frequency?
In general, i have noticed that i have simple accepted the fact that $$\psi = e^{i(kx-wt)}$$ represents a positive frequency, and $$\psi = e^{i(kx+wt)}$$ represents a negative frequency.
After a time ...
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Why is it always $\psi= A \sin kx + B \cos kx$ to solve a wave function?
Why is it always $\psi= A \sin kx + B \cos kx$ to solve a wave function instead of the one with $e^{ikx}$? Both are the solutions but the one with $e^{ikx}$ is seldom used.
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Relation $\Delta x \Delta p \approx \hbar$. Show that the condition $\Delta p \ll p$ guarantees that the packet does not spread
Consider a wave packet that satisfies the relation $\Delta x \Delta p \approx \hbar$. Show that the condition $\Delta p \ll p$ guarantees that the packet does not spread appreciably in the time it ...
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Does this relativistic generalization of the Schrodinger equation make sense? [duplicate]
So I'm aware that the correct relativistic approach to quantum mechanics is through quantum fields, but I'm still interested in the question that follows.
We know the Schrodinger equation in free ...
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1D bound state for a real potential
The prof says: "for 1Dimensional bound states with a real potential, the wave function is real, up to a phase".
The proof goes like this:
1D bound states are never degenerated. So $\Psi_{...
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How does $s$ subshell not have a node in the center despite the nucleus being there?
In most images of $1s$ subshell I see that there's no node shown at the center, and even the formula $n-\ell-1$ gives 0 as the answer.
But, isn't the nucleus experimentally proven to be at the center? ...
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Regarding Griffith quantum mechanics problem 2.47: Square double well
I have a query regarding part b) of the question. I do not understand in particular why $E_1$ and $E_2$ will vary as a function of $b$. With my understanding of the double rectangular potential ...
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Why does applying the kinetic energy operator to a free particle result in a divergent integral?
The wavefunction of a free particle is just
$$\psi = Ae^{i(kx-\omega t)}$$
and when you plug this into the Schrodinger equation you get the dispersion relation
$$E = \frac{\hbar^2 k^2}{2m}$$
However, ...
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How to derive Landau level with semiclassical approach?
I'm trying to derive the Landau level by applying semiclassical dynamics and the time-dependent Schrodinger equation. From that, I success to derive $E = \hbar\omega_c n$, but I fail to derive the ...
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Phase shift in square potential barrier when $E>V$
I'm trying to understand what happens to the phase of the wave reflected by a potential barrier when the energy $E$ is greater than the height of the barrier (i.e. $E>V0$) in the region $0<x<...
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Variational Method
A particle is moving in one dimension under a potential $V(x)$ such that, for large positive values of $x$, $V(x) \approx kx ^\beta$, where $k>0$ and $\beta$ $\geq$ 1. If the wave function in this ...
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Could one, in principle, make any predictions using the wavefunction of the universe? [closed]
Do physicists talk about the wavefunction of the universe? What does that wavefunction even mean? Usually, wavefunctions describe probabilities of measurements of a system. But in this case, every ...
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Why sum of squares of the magnitudes of Fourier coefficients in Infinite Square Well equals one but it is not so in regular Fourier analysis?
My question is basically this..
In regular math, Fourier Coefficients give the "amount" a particular frequency is available in any periodic signal. The squares of sum of coefficients is not ...
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Wave functions for two electrons in an infinite 2D potential well
Consider two electrons in a square 2D infinite potential well i.e $V=0\ for \ 0<x<a, 0<y<a, \ \ V=\infty$ everywhere else. Determine the energy and wavefunction(s) for the first excited ...
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Why not all Berry phase just vanished?
I just learned that for any real wavefuntions, berry phase equals zero. But in Griffiths' Problem 2.1(b), he proved that any complex wavefuntion can be written as linear combination of REAL ...
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Boundary Condition for Free Wave Function
In Secion 3.5 of Quantum Field Theory An Integrated Approach, Fredkin, the author talks about Aharanov-Bohm Effect, where it says
Define the wave function $$\Psi(\boldsymbol{r})=e^{i \theta(\...
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Ordinarily continuous function of the wave function
I just started studying quantum mechanics using the textbook Introduction to Quantum Mechanics by Griffith. Under the section of solving the Shrodinger equation for a Dirac delta potential, he ...
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How to evaluate a non-banal derivate?
I need to evaluate the following derivate:
$$\frac{dF}{d\Psi} = \frac{d}{d\Psi}\left[\beta\Delta\Psi+\alpha\left|\Psi\right|^2\Psi+\mu\Psi-i\vec{v}\cdot\bar{\nabla}\Psi\right]$$
where $\Psi$ is a ...
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A wave function normalized for a given time $t=0$ is normalized for every time $t \gt 0$ [closed]
Given $\Psi(x,t)$ a wave function such that
$$1=\int_{-\infty}^{\infty}\Psi^{*}(x,0)\Psi(x,0)dx$$
Prove that $\Psi(x,t)$ is normalized for every $t \gt 0$.
My approach on this has been the following:
...
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How to get the weight of an eigenstate inside the state of the system without knowing the state?
Let us suppose we have a system in a state $\Psi$, with:
$\Psi = \sum_m c_m \psi_m$
Let us further suppose that we don't know what $\Psi$ or the $c_m$ are, but that we know what the $\psi_m$ are since ...
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What is the physical meaning of the eigenstates of an operator in quantum mechanics?
Let us suppose that we have an Hamiltonian that describes a quantum system. If one would like to know all of the possible values that the energy of the system described by that hamiltonian, one has to ...
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How does the wavefunction of an antiparticle differ from that of the particle?
In this question I was answered that the invertion of wave function does not give antiparticles.
Then how does the wavefunction of an antiparticle look, given the wavefunction of the corresponding ...
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Degeneracy of wavefunction in 1 dimension
Suppose we have a one-dimensional bound state, with the degenerate eigenstates given by $\psi(x)$ and $\phi(x)$.
Using the Wronskian, we can show that there is no degeneracy, as the two functions are ...
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Energy (Hamiltonian) of Trial Wavefunction
Here I give a part of derivation of Hartree-Fock equations in case where basis functions (wavefunctions) are orthonormal and real: $$ \langle \psi_i | \psi_j \rangle = \langle \psi_j | \psi_i \rangle =...
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Definition of a wave packet
In Shankar's QM book page 168, the author stated
a wave packet is any wave function with reasonably well-defined
position and momentum.
What does he mean by resonably well-defined position and ...
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Uncertainty of waves
Here in the pictures I have written about some question I have been thinking about a long time, what do you think?
Link to the chapter I am talking about:
http://www.its.caltech.edu/~matilde/...
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Finding coefficients for wave function when Fourier transform is not possible
I am looking at a wave function moving towards a potential step with potential $V_0$ for $x>0$ while having a total energy that is smaller than $V_0$. I already know how you can find the unbound ...
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Superposition of momentum plane waves for a WF: discrete of continuous?
On the one hand there is a theorem that states that any reasonable wave function $\Psi$ can be written as a superposition of eigenstates of $\hat Q$ (a hermitian operator).
So if $\Psi _i$ are the ...
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Fourier-Transforming the Schrödinger Equation in order to solve it?
In Quantum all time favourites equation is given by: $$-\dfrac{\hbar^2}{2\,m}\,{\partial_{x}}^2\psi(x,t) = i\,\hbar\,\partial_t\,\psi(x,t)$$
What happens if you were to apply a Fourier-Transform on ...
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How do we predict the form of a wave function?
I’m seeing the same example of particle in a box all over. But it isn’t really clear how we know the form of a free particle, $\psi(x, t) = Ae^{i(kx - \omega t)}$
What if we had a wave function that ...
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How can we ignore the diverging term $e^\infty$ in the integral?
In Question (2.20) of Griffiths' Quantum Mechanics book, they have given this Solution.
In the Solution of question 2.20(b), they omitted $e^{(ik-a) \infty}$ (or may have considered $e^{(ik-a) \infty}=...
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What is the meaning of this wave function?
In these notes here the tight binding model for graphene is worked out.
The tight Binding Hamiltonian is the usual:
$$H=-t\sum_{\langle i,j\rangle}(a_{i}^{\dagger}b_{j}+h.c.)$$
where two different ...
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Schrödinger equation obtain $ψ(x,t)$ from $ψ(x,0)$
In this answer of the post "Wave packet expression and Fourier transforms" it is said that for the S.E. we have this property:
If we start with an initial profile $ψ(x,0)=e^{ikx}$, then the ...
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Radial Schrödinger equation: from $R_l(r)$ to $u_l(r)$
I am in the 3-dimensional radial Schrödinger equation, in the spherical coordinates, where we try to find the separable solutions
$$\psi(r) = R_l(r) Y_l^m(\theta, \varphi) \equiv \frac{u_l(r)}{r}Y_l^m(...
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What happens if the width $a$ of the potential well gets large or goes to infinity?
What happens to the wavefunction if for a 1D infinite potential well of certain width $a$, we let $a$ go-to infinity?
I think then is just a free particle and therefore it can be described with a ...
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Why massive and high energy massless objects tend to behave as particles?
I’m perfectly aware about the wave-particle duality of nature and therefore the difficulties of claiming what behaving like a particle is. This question is not about this very profound behavior of ...
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Time evolution of Gaussian packet
From Shankar's QM book pg. 154:
Consider the Gaussian wave packet at time $t=0:$
$$\psi(x',0)=e^{ip_0x'/\hbar} \frac{e^{-x'^2/2\Delta^2}}{(\pi\Delta^2)^{1/4}}.$$
Using the propagator $U(t)$ in the ...
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How would I normalize this ket vector? [closed]
So I am given the vector:
$$|Ψa⟩ = |x⟩ + |y⟩ − |z⟩$$
And I need to normalize it. I know that I have to take the dot product of the vector with itself (and it needs to equal 1) but how would I do this ...
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Can you explain the relationship between spin operator $S_z$ and rotation of a vector field (or wave function)?
I was studying Rotational Invariance in Quantum Mechanics from R. Shankar's book. Where I found on problem 12.5.1 that if $\vec{\Psi}(x,y)$ is a vector as such $\vec{\Psi}(x,y)=\psi_x(x,y)\hat{x}+\...
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How the concept of "wave function of atom" in Bose-Einstein condensate should be interpreted from perspective of quantum field theory?
A typical description of Bose-Einstein condensate goes along the line of "multiple atoms in the same ground state can be described by the same wave function". But hold on. Atoms are not ...
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Analogies for the Behavior of the Wave Function
In an overview of Quantum Field Theory, I recently heard the behavior of an elementary particle compared to that of a plucked (bowed) string. Although the stable states of excitation of the string are ...
