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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Significance of eigenvalues of an observable Of a wavefunction [closed]

What really is meant by eigenvalue of an observable? Does it mean that everytime we measure a value of an observable the result obtained is equal to the eigenvalue of the observable?
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503 views

Relation between wave number and wave vector? [closed]

What is the relation between wave number and wave vector? I've been self-studying wavefunctions as part of an intro into Quantum Mechanics. However, the textbooks, sites, and videos that I've seen do ...
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528 views

Is $e^{-2x}\sinh x$ an acceptable state wavefunction?

I have the following function in the range $(0, \infty)$: $$\psi(x)=e^{-2x}\sinh x$$ I would like to know if it is acceptable as a wavefunction. At $x = \infty$, we have $e^{-2x} = 0$ ...
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If we can say that the probability of finding an electron is 0.9 at point x, how does this tie in with the fact that an electron is a wave?

Shouldn't the electron be everywhere because it is a wave? Or am I wrong, an electron is not a wave? It just has an associated wave function that determines where it might be?
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290 views

Where is the info of the wavefunction stored at?

I have read these questions: Is the wavefunction a real physical wave or only a mathematical abstraction? The wavefunction gives you the probability description of a particle's position (and other ...
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116 views

Expectation of $r^k$ for a wave function [closed]

So I have a wave function in spherical polars: $$\psi=\left(r\sin(\theta)e^{i\phi}\right)^l\exp\left({\frac{-r}{(l+1)a}}\right)\;,$$ and I have to show $$E[r^k]=\frac{\int_0^\infty r^{2(l+1)+k}\exp\...
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367 views

Finite potential well problem when $ |E| \ll V_{0} $ [hold] [closed]

The potential well looks like the following. The problem requires the proof of the following formula in the case that $|E| \ll V_{0}$. E is the lowest energy eigenvalue. $$ \frac{\sqrt{...
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Why wave functions and not sums of vectors? [closed]

What are the benefits of representing quantum-mechanical states with wave functions rather than with sums of vectors?
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Why is this trial wave function a good choice to use with the variational principle?

I am learning about the variational principle and I want to understand how to pick "good" trial functions. For example: $$V(x) = \infty, x<0$$ $$V(x) = x, x \geq 0$$ and $$\psi (x) = 2\sqrt{\...
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695 views

Help normalizing a wave function

I don't see how the author normalizes $u(r)=Asin(kr)$. From Griffiths, Introduction to Quantum Mechanics, 2nd edition, page 141-142: My integral was $$\int_0^{\infty} \int_0^{\pi} \int_0^{2\pi}|A|^2 ...
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Integer number of wavelengths after the wavefunction is normalized in a cubic box

One of the steps when learning quantum mechanics in three dimensions is to normalize $\psi$ in a cubic box of side $a$. In math terms $$\int\limits_0^a\int\limits_0^a\int\limits_0^a \psi^*\psi dxdydz=...
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Is the Wave Function a Unitary Operator? [closed]

A unitary operator can be represented as an exponential $$e^{iA}$$ and as we represent the wave function in general as $$e^{i k x}.$$ Does that mean that the wavefunction is unitary as the exponent is ...
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Complex conjugate of hydrogen ground state wave function [closed]

For hydrogen atom ground state we know . I want to know the complex conjugate of .
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347 views

Infinite potential well question with wave function $\psi (x) = (x -a/2)^2$

In an infinite potential well with width $a$, a particle in this potential well is at state with wave function is $\psi (x) = (x -a/2)^2$ (not normalized). If you measure the energy of the system ...
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How can matter, the things which we can touch and feel on a macro level, exhibit a wave nature?

Louis de Broglie suggested that, if a particle like electron has momentum and wavelength associated with it (due to Planck's constant), then it might be a wave. The region where it exists are those ...
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45 views

Is there a minimum “Planck” probability analog of the minimum Planck distance? [closed]

I know the Planck distance is real, but is there a Planck probability? I know the wavefunction units is length to the negative 3/2 number of particles. So can we calculate a minimum probability from ...
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Normalisation of the following wavefunction: $\psi(\theta,\phi)=\cos(\theta)$ [closed]

Normalisation of the following wavefunction: $\psi(\theta,\phi)=\cos(\theta)$ So I thought about setting the following $N\int \cos(\theta)\cos^*(\theta) d\theta=1$ But then maybe I thought I was ...
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143 views

Discontinuous derivative of wavefunctions in the infinite square well potential problem?

I am intrigued about two points given in an answer to a similar question (https://physics.stackexchange.com/a/38198/262985). On one hand, it is stated that wavefunctions inside the well (excluding ...
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83 views

How to normalize this wave function? [closed]

My wave function is $$ \Psi = A e^\left({-\frac{\left|x\right|}{2a}- \frac{\left|y\right|}{2b} -\frac{\left|z\right|}{2c}}\right)dx $$ and I need to normalize it. I tried to take an integral of it and ...
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112 views

Why are numerical solutions for the Schrödinger equation necessary to plot this free waves solution?

Suppose a particle in free space given by: $$\psi(x,t) = Ae^{ik(x-\frac{\hbar k}{2m}t)} + Be^{-ik(x-\frac{\hbar k}{2m}t)}.$$ Why are numerical solutions necessary in order to plot this? Why can't ...
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Does entropy increase in an isolated quantum system undergouing unitary evolution?

Does the second law of thermodynamics still work without wavefunction collapse? I received these contradicting answers on Quora: https://qr.ae/TWvoOC https://qr.ae/TWvoO4
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How did scientist get photos of wave function of electron in the double slit experiment? [closed]

In the double slit experiment I know that the electron fires as a particle one at a time then splits goes through both slits and recombined and interferes with itself and hit the wall creating a ...
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Infinite square well: what's the intuition behind there being no chance to measure energy in states where n is even?

In an infinite square well of width $a$ (running from 0 to $a$), the wavefunction is, $$|\psi_n\rangle=\psi_n(x)=\sqrt{\frac{2}{a}}\sin\left(\frac{n\pi}{a}x\right)$$ What's the intuition behind there ...
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591 views

Write the dimension of 1D wave function? [closed]

I want to know how to find the dimension or unit of one-dimensional wave function
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Considering an arbitrary wavefunction for a free particle, are all normalizable functions valid?

One can show that a possible solution to a wavefunction with constantly zero potential is equal to, only considering the spacial piece: $$\psi(x) = \int_{-\infty}^{\infty} A(k) \ e^{ikx} dk$$ This ...
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Probability of finding a particle in the solid angle $d\Omega$ at $\theta$ and $\phi$ [closed]

For a spinless particle with the wavefunction \begin{equation} \psi(x,y,z)= K(x+y+2z)\exp(-\alpha r) \end{equation} with $r=\sqrt{x^2+y^2+z^2}$ and K and $\alpha$ are real constants. I have to ...
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Eigenvalue problem in Hilbert Space

In page 71 of Shankar's Principles of Quantum Mechanics, the author states the following(kindly take a look at page 70 because the following is a part of an example problem): The allowed ...
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How can $e^{-i\omega t}$ be simplified when normalizing a wave function? [closed]

I'm going through some of the practice problems in a QM textbook. (This isn't a homework question, I'm not taking a class.) When normalizing the wavefunction $$\Psi= Ae^{-\lambda |x|}e^{-i\omega t}$$ ...
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Dirac Delta potential

As we know a particle in attractive Dirac delta potential has discontinuity in the derivative of its wavefunction. I have two questions in this regard: Can a second order differential equation be ...
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How to prove an expression is not a solution in 1D Schrödinger equation? [closed]

I was reading Griffith's book Introduction to Quantum mechanics and found that for the case of a free particle, we can diregard solutions of the form $e^{kx}$, where $k$ is real (positive or negative)....
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How to solve Schrodinger equation back in time to find past wavefunction from which present wavefunction has been evolved?

How to solve Schrödinger equation back in time to find past wavefunction from which present wavefunction has been evolved? i.e. Suppose, at present or at this moment I know $\psi_{present}(r)$. ...
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Psi Ontic interpretations of QM that can be made relativistic?

What Psi Ontic interpretations of QM can be made relativistic? Is the Many Worlds interpretation the only one?
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820 views

A Problem on Schrodinger's Equation and wave functions [closed]

QUESTION: Let $\Psi(x,t) = F(x).G(t)$ be a solution of the time-dependent S.E. Then show that $F(x)$ satisfies the time-independent S.E. MY ATTEMPT: The time-dependent S.E is given by: $$i\hbar\...
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Is a free particle describable by wave packet?

A free particle wave function can be represented as $Ae^{ikx}$, where $A$ is some constant. My understanding on the topic is free particles cannot be normalized. That is why we have to use box ...
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493 views

Schrodinger equation: free particle solution in integral form? [closed]

In one exercise I was asked to find the general solution for a free particle but in integral form(!!) I'm supposed to get but I have no idea how to get there! Looking around I found quite a bit of ...
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843 views

Test charge and source charge test for EM field strength

We know how to determine the strength/density of an electric/magnetic field by placing a test charge in the source charge's electric field and then measuring the force the test charge experiences. How ...
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Could someone explain this paragraph?

"Although Arnold Sommerfeld’s theories had used quantum mechanics, his focus had been on the probability of the electron distribution. The phases of the individual wavefunctions had been ignored. ...
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How do we compute the expectation value of momentum for a 1D wave function?

I have a 1D wave function $\psi(x)=Ae^{-x^2/a^2}$ and $a$ is just a value of $x$ greater than $0$ and $A$ is the normalization constant, which I found to be $A= \frac{1}{\sqrt{a \, \sqrt{(\pi/2)}}}$. ...
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Outcome of measuring $L_x$ of a linear combination of hydrogen states

I have encountered a question: What are the possible outcomes in measuring $L_{x}$ and what are the corresponding probability of the state: $\Psi(r,0)=1/2(\Psi_{200}+\Psi_{310}+\Psi_{311}+\Psi_{31-...
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Bound states of Dirac Delta function in infinite well

If there is a potential of $-\alpha\delta(x)$ for $-a<x<a$ and $\infty$ elsewhere, and the energy of the system is less than 0, then I'm trying to find the wave function. From the Schrodinger ...
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Probability of finding particle in infinite square well, displaced walls

Initially a quantum particle moves in a one-dimensional well ($x$-axis) from $-a$ to $ a$, $ V = \infty $ outside and $ V = 0 $ inside the well. So initially, the wave-function $$ \psi_0 = \sqrt\frac{...
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Quantum mechanics: Finite square well problem

What will happen if the potential is less than 0, for instance $V(x)=-10eV$. Is this means there will be no bound states? Since solution to the time independent Schrodinger equation (those discrete ...
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A particle starts out in a linear combination of two stationary states [closed]

I have a doubt in finding the probability density. I would be thankful if someone helps me out in this question: $E_1$ and $E_2$ are the energies associated with $\Psi_1$ and $\Psi_2$. It follows ...
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4-momentum operator, in geometric algebra [closed]

David Hestenes suggests that the relativistic wavefunction can be expressed in geometric algebra as follows: $$ \psi = \rho^{1/2}e^{B/2I} R \tag{2} $$ where $$ \psi \tilde{\psi}=\rho e^{B/2I} e^{-B/...
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Does the Schrodinger equation obey the rule for differentiating a function if the function is in terms of the wavefunction?

Does the Hamiltonian operator act like a derivative when acting on a functional in terms of wavefunctions? For example, does $$H\psi^2=2\psi H\psi$$ hold true? More generally, if the functional, $F(\...
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Quantum Mechanics - Derivation of the fact that determinate states are eigenvectors of the $\hat{Q}$ operator

In the Intro to Quantum Mechanics textbook by Griffiths on page 105 equation [3.116], I'm confused about 3 things. What is $$\hat{Q}$$ supposed to represent? At first they use the variance definition ...
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2answers
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Are there exactly three physically relevant operators that commute with $H$ for helium?

I am thinking about something I learnt as an undergraduate. In the section on identical particles (page 212) of Griffiths book on quantum mechanics he speaks of helium and says that: The excited ...
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Quantum mechanical measurement

Suppose, a particle has non zero probability to be in any between x = 1 and x = 10. Let, we measure its position with a very low energy photon. It will collapse it’s wave function, since it disturbs ...
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Particles' Wave Function Capability

Do all particles have the ability to have a wave state even if they are a part of objects larger than those seen tested in the double slit experiment? Do all particles have the ability to have a ...