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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Why is the reflection coefficient in quantum mechanical scattering defined this way?

In Griffiths' "Introduction to Quantum Mechanics, second edition" section 2.5.2, p. 73, he states: For the delta-function potential, when considering the scattered states (with $E > 0$), we have ...
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61 views

Should state vectors be considered constant?

By the principle of superposition, a state vector can be defined as $$\begin{align} \psi(x) &= c_1 \psi_1(x) + c_2 \psi_2(x) + \cdots + c_n \psi_n(x) \\ \lvert\psi\rangle &= \begin{pmatrix}c_1 ...
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28 views

How to tell when exponentials are real valued? (Barrier Potential)

Where $k_1=\frac{\sqrt{2mE}}{\hbar}$ and $\alpha=\frac{\sqrt{2m(V_0-E)}}{\hbar}$ I'm quite confused as to why the exponentials in regions I and III are complex functions while in region II the ...
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39 views

Existence of representation of symmetry transformation

There is a simple fact that we can change our point of view and that physical laws should remain the same, id est, outcomes of our experiments should be the same no matter from which frame of ...
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24 views

Perodic boundary conditions vs Dirichlet?

I have been working through several examples recently involving particles in boxes (when finding the partition function of an ideal gas for example or looking at photon gases). I have seen two ...
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44 views

Aharonov-Bohm effect [closed]

In the build up to Aharonov-Bohm effect, one has to represent the gauge covariant form of STIE. We need to consider two things; the vector potential A changes`under gauge transformation and as a ...
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137 views

What are the functions of these coefficients $c_1,c_2,c_3,c_4$ in $ \psi_{sp^3}= c_1\psi_{2s}+ c_2\psi_{2p_{x}} + c_3\psi_{2p_y}+ c_4\psi_{2p_{z}}$?

Hybridised orbitals are linear combinations of atomic orbitals of same or nearly-same energies. Atomic orbitals interfere constructively or destructively to give rise to a new orbital which is what we ...
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99 views

Difference between a wavevector and wavefunction

I often see both terms used in textbooks, but I am not sure whether I understand the difference between them. Both describe the state of a system, however, they seem different in some ways. From what ...
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59 views

General expectation value

I have a basic question related to finding expectation values of an operator $\hat{Q}$ We know that the expectation of $\hat{Q}$ (in the position space) is given by $$\langle Q \rangle=\int {\Psi^* ...
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112 views

Derivation of Schrodinger's wave equation

To derive $$i \hbar \frac{\partial}{\partial t} \psi = H \psi,$$ we start with $$i \hbar \frac{\partial}{\partial t} |\alpha \rangle = H| \alpha \rangle$$ and then multiply by $\langle x|$ on the ...
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68 views

Perturbation by electrical field in infinite potential well: difference in first energy corrections because of difference in the limits of the well

In time independent perturbation theory we can calculate the first and second energy corrections resulted by a potential V in the Hamiltonian $ H=H_o + λV , $ , λ<<, by the expressions: $$ε_1 = ...
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201 views

Does measurement change the evolution of wave function?

Basically any measurement is on wave function $|\psi\rangle$ is done by operator $X$ such that $X|\psi\rangle$ results observable $x$ with some probability. But what happens to $|\psi\rangle$? Does ...
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687 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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49 views

Probability of finding electron in a spherical shell

In the book Arthur Beiser - Concepts of modern physics, probability of finding an electron in Hydrogen atom in the spherical shell between $r$ and $r+dr$ is given as \begin{equation} P(r)dr = r^2|R(r)...
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42 views

Continuous spectra and quantum decoherence

Suppose that some quantum wave function $\psi = \int a_i \,i\rangle \,\,di$ where pseudo-spanning ket "vectors" of $\psi$, $i \rangle$, are continuous. (thus the use of integral.) By normalization, $...
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150 views

Minimum uncertainity

I'm confused in finding the condition for minimum uncertainty, The author in the book I refer goes on saying that $|g\rangle=c|f\rangle$ is the condition for minimum uncertainity for some constant $...
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48 views

Understanding wave functions of matter waves

The wave functions of matter waves give the probability density of the particle being at a certain location. Does this arise because as an outside observer, we have incomplete information about the ...
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90 views

Momentum and position for free particle

In the section of 'The free particle' in 'Introduction to quantum mechanics, second edition' by Griffiths page 65. He has the wave equation as $$\Psi(x,t) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\...
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458 views

Probability current vs. direction of wave function

I did an exercise for my Quantum-Mechanics Lecture: Let $\hbar$=2m=1. A particle in 1 dimension has $j(x)=2\ Im(\overline{\psi} (x) \ \psi'(x))$ and it's to show that there are superpositions $\psi (x)...
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161 views

Where are the worlds in many-worlds interpretation?

What does it mean in MWI for other universes to exist? Are they in some sector of spacetime beyond our cosmic horizon or is it more complicated? I'm not asking this on Philosophy SE because people ...
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196 views

Ground state of a particle in a ring - angular momentum is 0, but it is 'rotating' anyway?

Particle in 1D ring is a textbook problem, but there is one thing I don't understand - if the ground state is considered to have zero angular momentum, then its energy is also zero. And the ...
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Bloch Functions as an implication of the Crystallographic Restriction Theorem?

I'm studying Bloch Functions and it seems to me safe to assume that they are the most general Eigenfunction of a Hamiltionian with the crystal periodicity. Now the only considerations made in deriving ...
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95 views

The time evolution in Dirac delta potential [closed]

We know that the dirac delta potential has exactly one bound state. If the potential strength suddenly changes a value, the bound state should evolve to the new bound state, how to describe the time ...
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Is the sum of two stationary states of different energies also a stationary state?

The question title kind of speaks for itself really. I was thinking of maybe using the orthogonality relation to try to show this: $$\int_{-\infty}^{\infty}\phi_n(x)\phi_m(x)dx=\delta_{nm}.$$ ...
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Does an expectation value over only part of a wavefunction have physical meaning?

Does the expression: $$\langle p\rangle_{x=a, b} =\frac{\int_a^b \Psi(x)^*\,\hat{p}\,\Psi(x)dx}{\int_a^b |\Psi(x)|^2dx} $$ have any physical meaning when $\int_a^b |\Psi(x)|^2dx\neq\int_{-\infty}^{\...
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A quantum particle which is almost at rest but whose position is random!

Assume a particle is given by a quantum state which is constructed in such a way that it is equally probable to find it anywhere in an fixed interval $(0,L)$ but has arbitrarily low velocity. The ...
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Complex Conjugate of Wave Function

I've been reading through Griffiths QM book, and the only thing bugging me is they never fully described what $\Psi^* $ should be for any given function. I know it's the complex conjugate at the same ...
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63 views

Loss of interference in single-photon Mach–Zehnder interferometer with detector in only one arm

I have read that if you have a Mach–Zehnder interferometer (doing a single-photon experiment) and put a non-destructive detector in only one of the two arms (connected to the first beam splitter), you ...
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124 views

Help needed to understand “On the reality of the quantum state”

I am having trouble to understand the reasoning in the following paper, On the reality of the quantum state. MF Pusey, J Barret and T Rudolph. Nature Phys. 8, 475–478 (2012); arXiv:1111.3328. ...
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218 views

Single quantum particle in beam splitter, with different systems located in each channel

Suppose a quantum mechanical particle enters a beam-splitter, which sends its wave packets into two mutually orthogonal channels, $C_a$ and $C_b$. Suppose that $C_a$ also contains System A, with ...
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29 views

Deriving Wave Function for Scattering States with Delta-Function Potential

I am following the Griffiths Book on Quantum Mechanics, and am following the derivation for the wave function for Delta-Function Potentials. $$V(x) = -\alpha \delta(x)$$ In the scattering states, ...
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How can I solve this quantum mechanical “paradox”?

Let a (free) particle move in $[0,a]$ with cyclic boundary condition $\psi(0)=\psi(a)$. The solution of the Schrödinger-equation can be put in the form of a plane wave. In this state the standard ...
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488 views

How to determine the transmission coefficient of a gaussian wave packet scattering on an finite square well?

I am doing a scattering simulation of a Gaussian wave packet on a finite square well. I have solved numerically the Schroedinger equation and I know the values of the wave function after the ...
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Is there a “natural” way to interpolate between a set of bound state wave functions?

Consider for example the Coulomb potential, $-Z/r$, for which there exist a set of bound states with energy $\epsilon_n := {-Z^2 \over 2 n^2}$ (in Hartree). If I want the "wavefunctions" for some ...
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Why does the wave description say that probability oscillates, while the phase interpretation says constant amplitude?

The wave description of a particle illustrates an oscillating probability of the particle being found in any point in space. When a particle travels, it carries along with it a phase that oscillates ...
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418 views

Normalized wave functions in position and momentum space

Using the following expression for the Dirac delta function: $$\delta(k-k')=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(k-k')x}\mathrm{d}x$$ show that if $\Psi(x,t)$ is normalized at time $t=0$, ...
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68 views

What's the microscopic and macroscopic effect of wavefunction dispersion?

In Quantum Mechanics (Merzbacher 2nd ed.), problem 2.1 asks us to derive the time evolution of a one-dimensional Gaussian wavefunction (formula given for $t=0$), assuming the velocity is in the $+x$ ...
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Notation of complex valued atomic orbitals

Atomic orbitals are usually labeled $1s$, $2p_x$, $2p_x$, $2p_z$ and so on. These wave functions are defined to be real valued. The original wave functions are complex valued. The $2p_x$ orbital is ...
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58 views

Expectation value of the Hamiltonian [closed]

How to calculate expectation value of the Hamiltonian for hydrogen atom? $$\langle H \rangle_{\alpha l} \equiv \frac{\langle \psi_{\alpha l m}|H(r)| \psi_{\alpha l m}\rangle} {\langle \psi_{\alpha l ...
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How to rewrite a wave function in terms of spheical harmonics

I'm given a wave function for a particle, in three variables (spherical coordinates): $ψ=ψ(r, θ, φ) = re^{-r/a}sin(θ)sin(φ)$. I'm tasked with rewriting $ψ$ in terms of spherical harmonics which are ...
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1answer
69 views

Energy of hydrogen atom - Schrodinger equation [closed]

The wavefunction of the electron in the hydrogen atom is $ k* exp(-r/a)$ (k is the normalization constant), but it does not take n into account, whereas the solution of Schrödinger's equation ($H(...
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1answer
91 views

Clarification about two forms of the wave function

The wave function in the position representation is $\langle\ x\rvert\psi\rangle$ = $ \psi (x) $ , where $ \psi (x) $ are the continuous coefficients that multiply the orthonormal basis vectors, i.e, $...
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2answers
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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 ...
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98 views

Difference for boundary condition, particle in a box

When solving the simple problem of a free particle in a box of volume $V = L^3$, we can impose either periodic boundary conditions $\psi(0) = \psi(L)$ and $\psi '(0)= \psi'(L)$ either strict boundary ...
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235 views

Hartle and Hawking's universal wavefunction

My apologies in advance if this question is poorly worded or doesn't make any sense, however I have just finished reading into this theory and it seems as though Hawkings No Boundary Universe is ...
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1answer
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How to calculate the frequency of oscillation of superposition states [closed]

Been working on this question for a while and I'm not sure how to go about it. Could someone point me in the right direction, particularly for the frequency question. The question is as follows: A ...
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4answers
865 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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Why wavefunction is sometimes multiplied by the radius to get probability density?

When solving 1d particle in a box, the probability density is said to be proportional to $|\psi|$, but when solving 3d orbitals, the probability density is said to be proportional to $|\psi|^2 r^2$. ...
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62 views

Two fermions with total spin 1 antisymmetric wave function? [closed]

How can I prove, that two fermions with a total spin of 1 must have an antisymmetric wave function?
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55 views

Can we measure the energy of one of several identical particles?

Suppose we have a many-particle system described via a many-particle wavefunction that involves single-particle states $\lvert\lambda_{a}\rangle$, $\lvert\lambda_{b}\rangle$, $\lvert\lambda_{c}\rangle$...