I am unable to complete the first part of the question. After I plug in the function for psi into the differential equation I am stuck:
$$\frac {d \psi (r)}{dr} = -\frac 1 a_0 \psi (r), \frac d{dr} \biggl(r^2 \frac {d\psi (r)}{dr} \biggr) = -\frac 1 {a_0}\frac d {dr} \bigl[r^2 \psi(r) \bigr] =...
I know how to solve this problem when the energy at ground state is zero but I don't know how to deal with 1st excited state energy as zero.
According to me since the potential energy is zero therefore the kinetic energy must be 13.6eV according to conservation of energy.
I also know that the...
I can not solve this problem:
However, I have a similar problem with proper solution:
Can you please guide me to solve my question? I am not being able to relate Y R (from first question) and U (from second question), and solve the question at the top above...
hi guys
i am having a little problem concerning the theta part of TISE :
its clearly that its very similer to the associated Legendre function :
how iam going to change 1/sinθ .... to (1-x^2) in which x = cosθ
i tried many identities but i am stuck here .
any help on that ?
Hello, I have a little problem understanding the quantum mechanics of a hydrogen atom.
Im troubled with the following question: before i measure the state of a (simplified: without fine-, hyperfinestructure) hydrogen atom, which is the right probability density of finding the electron? is it...
While separating variables in the Schrodinger Equation for hydrogen atom, why are we taking separation constant to be l(l+1) instead of just l^2 or -l^2, is it just to make the angular equation in the form of Associated Legendre Equation or is there a deeper meaning to it?
Homework Statement
Real atomic nuclei are not point charges, but can be approximated as a spherical distribution with radius ##R##, giving the potential
$$ \phi(r) = \begin{cases}
\frac{Ze}{R}(\frac{3}{2}-\frac{1}{2}\frac{r^2}{R^2}) &\quad r<R\\
\frac{Ze}{r} &\quad r>R \\...
Homework Statement
Assume that Planck's constant is not actually constant, but is a slowly varying function of time, $$\hbar \rightarrow \hbar (t)$$ with $$\hbar (t) = \hbar_0 e^{- \lambda t}$$ Where ##\hbar_0## is the value of ##\hbar## at ##t = 0##. Consider the Hydrogen atom in this case...
Hi, I have a question about calculating probabilities in situations where a particle experiences a sudden change in potential, in the case where both potentials are time independent.
For example, a tritium atom undergoing spontaneous beta decay, and turning into a Helium-3 ion. The orbital...
Homework Statement
How to calculate the probability of finding an 1s electron within 1 picometer cubic region located 50pm from the nucleus.
Homework Equations
The probability of an 1s electron within a spherical volume of radius 'a' from nucleus can be find using the expression...
Hi everyone;
A very stupid confusion here. When we want to talk about the most probable radius to find the electron in $1s$ orbital, why do we talk about the radial density and not the probability itself? For instance, the probability of finding the the electron at a radial distance $r$...
I have to do a Taylor expansion of the energy levels of Dirac's equation with a coulombian potential in orders of (αZ/n)^2 , but the derivatives I get are just too large, I guess there is another approach maybe?
This is the expression of the energy levels
And i know it has to end like this:
Homework Statement
I am having trouble with part d, where they ask me to prove that the wave function is already normalized
The Attempt at a Solution
But that clearly doesn't give me 1. I tried to use spherical coordinates since it is in 3D? Not really sure how to proceed.
EDIT: I realize...
I'm trying to prove that the wave function of Hydrogen for the fundamental state is normalized:
$$ \Psi_{1s}(r)=\frac{1}{\sqrt{\pi a^3}}e^{-\frac{r}{a}} $$
What I tried is this:
$$ I= \int_{-\infty}^{\infty} | \Psi^2(x) | dx = 1$$
$$ \int_{-\infty}^{\infty} \frac{1}{\pi...
Homework Statement
A Hydrogen atom is interacting with an EM plane wave with vector potential
$$\bar A(r,t)=A_0\hat e e^{i(\bar k \cdot \bar r -\omega t)} + c.c.$$
The perurbation to the Hamiltonian can be written considering the proton and electron separately as...
Today I was doing some reading and I came across this topic. If we have a stationary hydrogen atom with a single electron in orbit around the nucleus and want to calculate the kinetic energy of the electron we would take the following approach.
1) Using Newton's second law:
F = ma ⇒ FE = mac...
Homework Statement
This is a (long) multi-part question working through the various stages of solving the radial Schrodinger equation and as such it would be impractical to type it all out here but I will upload the pdf (https://drive.google.com/open?id=0BwiADXXgAYUHOTNrZm16NHlibUU) of the...
Homework Statement
Essentially we are describing the ODE for the radial function in quantum mechanics and in the derivation a substitution of u(r) = rR(r) is made, the problem then asks you to show that {(1/r^2)(d/dr(r^2(dR/dr))) = 1/r(d^(2)u/dr^2)
Homework Equations
The substitution: u(r) =...
I have some confusions, and I would like some help:
What states will hydrogen atom be before it emits a photon?
Will it possible be superposition of its eigenstates? (If so, then by measuring the energy of photon, we measure its' energy causing its wavefunction collapse, am I right?)...
Homework Statement
Suppose we have a wavefunction with n=4. If we measure the orbital angular momentum along the z-direction(no spin in this problem) and get 2*hbar then what are the possible values of the total angular momentum and what is the most general wavefunction after the measurement...
Homework Statement
We model the Hydrogen atom as a charge distribution in which the proton (a point charge) is surrounded by negative charge with the volume density of ρ = -ρ0 * exp (-2r/a0) where a0 is the Bohr radius. And ρ0 is a constant chosen such that the entire atomic distribution is...
I am trying to calculate the Lyman-alpha wavelengths of photons emitted from different hydrogen-like atoms such as deuterium and positive helium ion 4He+, using the relation 1/λ = R*|1/ni^2 - 1/nf^2|, where R is the Rydberg constant and ni and nf are integer numbers corresponding to the initial...
I'm doing a homework problem where it asks to calculate the diameter of a hydrogen atom with n=600. I used the equation $$r=\frac{n^2a_0}{Z}$$ where $$a_0=0.529e^{-10}m$$.
Solving for r yields:
$$r=\frac{(600^2)(0.529e^{-10}m)}{1}=1.90e^{-5}m$$
Multiplying by 2 to get the diameter yields...
Hello Friends !!
I have a question regarding binding energy...
Trying to calculate the binding energy of H-1 (hydrogen nucleus).
Well it is obvious that the binding energy is zero since there is no other nucleons that the proton is bound to.
But after having collected the best possible data of...
Suppose a single hydrogen atom is in mixed state.
Ψ=(1/√2) Ψ_100+(1/√2) Ψ_200
Then energy will be E=(1/2)*13.6+(1/2)*(3.4)=8.5 eV.
But there is no spectral line at 8.5 eV.
Consider the Dirac equation for bounded electron in hydrogen atom.
I am trying to get a clear physical explanation for all mathematical terms that appear in the Hamiltonian and energy spectrum.
Kinetic and Coulombic potential and rest energies are the first terms and easy to identify.
Then we...
Homework Statement
Show that in terms of the dimensionless variable ##\xi## the radial equation becomes ##\frac{\mathrm{d}^{2} u}{\mathrm{d} \xi^{2}}=(\frac{l(l+1)}{\xi^{2}}-\frac{2}{\xi}-K)u##
Homework Equations
##u(r)\equiv rR(r)##
##\xi \equiv \sqrt{2\mu U_{0}}\frac{r}{\hbar}##...
Homework Statement
"Suppose that a hydrogen atom, initially in its ground state, is placed in an oscillating electric field ##\mathcal{E}_0 \cos(\omega t) \mathbf{\hat{z}}##, with ##\hbar \omega \gg -13.6\text{eV}##. Calculate the rate of transitions to the continuum."
Homework Equations
##R =...
Hi guys,
I consider the qm-derivation of the electronic states of hydrogen.
There are two different derivations (I consider only the coulomb-force):
1) the proton is very heavy, so one can neglect the movement
2) the proton moves a little bit, so one uses the relative mass ##\mu##
The...