Serial of year 27

• Any theory of quantum gravity is useful only when we deal with very small distances where the effects of gravitation are comparable to quantum effects. Gravitation is characterized by the gravitational constant, quantum mechanics by the Planck constant, and special relativity by the speed of light. Look up numerical values of these constants, and, using standard algebraic operations, combine them to obtain a quantity with the dimensions of length. This is the length scale where both quantum mechanics and gravitation are important.

• Prove that the special Lorentz transform (i.e. a change of the reference frame to one that is moving with speed $v$ in the $x¹;$ direction)

$$x^0_\;\mathrm{nov}=\frac{x^0-\frac{v}{c}x^1}{\sqrt{1-\(\frac{v}{c}\)^2}}\,,\quad x^1_\mathrm{nov}=\frac{-\frac{v}{c}x^0 x^1}{\sqrt{1-\(\frac{v}{c}\)^2}}\,,\quad x^2_\mathrm{nov}= x^2\,,\quad x^3_\mathrm{nov}= x^3$$ leaves the spacetime interval invariant. * Set $Δx=Δx=0$ in the definition of a spacetime interval. You should get

$$(\Delta s)^2 = -\(\Delta x^0\)^2 \(\Delta x^1\)^2$$

What is the region of the plane ( $Δx^{0},Δx¹;)$ where the spacetime interval ( $Δs)$ is positive? Where negative? What is the curve ( $Δs)=0?$

• What are the physical dimensions of action? (What are its units?) Does it have the same unit as one of fundamental constants from the first question in the previous part of the series? Which one?

• $ From Niels Bohr$ – Assume the motion of a point mass on a circle with the centripetal force of

$$F_\;\mathrm{d} = m a_\mathrm{d} = \frac{\alpha}{r^2}\,,$$

where $ris$ the radius of a circle and $α$ is some constant. Then

• Calculate the reducted action $S_{0}$ for one revolution as a function of its radius $r$.
• Determine the values of $r_{n}$, for which the value of $S_{0}$ is merely the constant from the sub-task a) multiplied by a natural number.
• The total energy of the point mass is $E=T+V$. For this force it istrue that $V=-α⁄r$. Express the energy $E_{n}$ of the particles depending on the radii $r_{n}$ using said constants.

Tip Youshould have encountered radial motion in your high-school education and also the relationships between displacement, velocity and acceleration. Use them and then the integration of action along the circumference of the circle with a constant $r$ shall become easier (constant quanties can be easily factored out of the integral). Don't forget that the path integral of „nothing“ is merely the length of the integrated path.

• The last sub-problem may seem complicated but it is merely a excercise in differentiating and integrating simple functions. You should be able to do it nly with standard table integrals and derivatives. Show that the full action $S$ for a free particle moving from the point [ 0$;0]$ to the point [ 2$;1]is$ for the case of linear motion (first case) minimal. In other words that it is bigger in the two other cases

$$\mathbf{y}(t)&=\left(2t,t\right) \,,\\\mathbf{y}(t)&=\left(1-\cos{(\pi t)} \frac{1}{\pi}\sin{2\pi t}, t\right) \,,\\\mathbf{y}(t)&=\left(2t, \frac{\;\mathrm{e}^t-1 t^2(t-1)}{\mathrm{e}-1}\right) \,,$$

where e is the Euler number. Tip First find the derivative of $\textbf{y}(t)$, put it into the equation for action and integrate.

• In the text of the seriesy we used the approximative relation √( 1 + $h)$, where $his$ a small value. Determine the precision of such an approximation. How much can $h$ differ from zero so that the approximated value and the precise one shall differ only by 10%? We can make a similar approximation for any „normal“ (read occuring in nature) function using Taylor's series expansion. Try to find the Tylor's series of cos$h$ and sin$h$ on the internet and neglect factors with a higher order than $h$ and find the approximate border value where it differs by approximately 0.1.

• Considering a wave equation for a classical string from the serial and let the string be fastened on one end in the point [ $x;y]=[0;0]$ a na druhém konci v bodě [ $x;y]=[l;0]$. For what values of $ω,α,aabis$ the expression

$$y(x,t)=\sin ({\alpha} x)\left [a\sin {({\omega} t)} b\cos {({\omega} t)}\right ]$$

a solution of the wave equation? Tip Subsitute into the equation for motion and use the boundary conditions.

• In the previous part of the series we were comparing different values of action for different trajectories of different particles. Now calculate the value of Nambu-Gota's action for a closed string which from time 0 to time $t$ stands still un the plane ( $x¹,x)$ and has the shape of a circle with radius $R$. Thus we have

$$X({\tau} , {\sigma} )=(c{\tau} , R\cos {{\sigma} }, R\sin {{\sigma} },0)$$

for $τ∈\langle0,2π\rangle$. Furthermore sketch the worldsheet of this string (forget about the last zero component) and how the line of a constant $τ$ and $σ$ look.

• Look into the text to see how the operator of position $<img$

src=„https://latex.codecogs.com/gif.latex?\hat%20X“>$

and momentum $<img$ src=„https://latex.codecogs.com/gif.latex?\hat

%20P“>$ acts on the components of the state vector in $x-$

representation (wave function) and calculate their comutator, in other

words

<img src=„https://latex.codecogs.com/gif.latex?(\hat%20{X})_x%20\left((\hat%20

{P})_x%20{\psi}%20(x)\right)%20-%20(\hat%20{P})_x%20\left((\hat%20{X})_x%20

{\psi}%20(x)\right)%20“>

Tip Find out what happens when you take the derivative

of two functions multiplied together

• The problem of levels of energy for a free quantum particle in other words

for $V(x)=0$ has the

following form:

<img src=„https://latex.codecogs.com/gif.latex?-\frac%20{\hbar%20^2}

{2m}%20\dfrac{\partial^2%20{\psi}%20(x)}{\partial%20x^2}=%20E%20{\psi}%20

(x)\,.“>

• Try inputting $ψ$

( $x)=e^{αx}$ as the solution

and find out for what $α$ (a general complex number)

is $Epositive$ (only use such $α$ from now on).

• Is this solution periodic? If yes then with what spatial period

(wavelength)?

• Is the gained wave function the eigenvector of the operator of momentum

(in the $x-representation)?$ If yes find the relation between

wavelength and momentum (in other words the respective eigenvalue) of the state.

• Try to formally calculate the density of probability oof presence of the

particle in space.naší vlnové funkci podle vzorce uvedeného v textu. Pravděpodobnost, že se

částice vyskytuje v celém prostoru by měla být pro fyzikální hustotu pravděpodobnosti 1,

tj. <img src=„https://latex.codecogs.com/gif.latex?\int_\mathbb{R}%20\rho

(x)%20\mathrm{d}%20x=1.“> Show that our wave function can't be

$normalized$ (in other words multiply by some constant) so that its formal

density of probability according to the equation from the text was a real

physical density of probability.

• *Bonus:** What do you think that the limit of the

uncertainity of a position of a particle is if the wave function it has is close

to ours (In other words it approaches it in all properties but it always has a

normalized probability density and thus is a physical state) Can we (using Heisenberg's relation of uncertainty) determine what is

the lowest possible imprecision while finding the momentum?

Tip Take care when dealing with complex numbers. For

example the square of a complex number is different than that of its magnitude.

• In the second part of the series we derivated the energy levels of an

electron in hydrogen using reduced action. Due to a random happenstance the

solution of the spectrum of the hamiltonian in a coulombic potential of a

proton would lead to thecompletely same energy,in other words

<img src=„https://latex.codecogs.com/gif.latex?E_n%20=%20-{\mathrm{Ry}}%20\frac

%20{1}{n^2}“>

where Ty = 13,6 eV is an ernergy constant that is known

as the Rydberg constant. An electron which falls from a random energy

level to $n=2$ shall emit energy in the form of a proton

and the magnitude of the energy shall be equal to the diference of the energies

of the two states. Which are the states that an electron can fall from so that

the light will be in the visible spectrum? What will the color of the spectral

lines be?

Tip Remember the photoelectric

effect and the relation between the frequency of light and its

wavelength.

• We consider only open strings and we shall limit ourselves merely to three dimensions. Draw how the following things look like

• a string moving freely through timespace,
• a string fixed with both ends to a D2-brane,
• a string between a D2-brane and D1-brane.

Where can the strings end in the case of three parallel D2-branes?

• Choose one of the functions

$$\mathcal{P}_{\mu}^{\tau}$$

ot $$\mathcal{P}_{\mu}^{\sigma}$$ that was defined in the first part of the series and find its explicit

form (in other words a direct dependence on $$\dot{X}^{\mu}$$ and <img

src=„https://latex.codecogs.com/gif.latex?X'^{\mu}“>). Show that the conditions $$\vect{X}'\cdot \dot{\vect{X}}=0$$

and $$|\dot{\vect{X}}|^2=-|\vect{X}'|^2$$

• Find the spectrum of energies of a harmonic oscilator.

• The energy of the oscilator is given by the hamiltonian

$$\hat{H}=\frac{\hat{p}^2}{2m} \frac{1}{2}m\omega^2\hat{x}^2$$

The second expression is clearly the potential energy while the first gives after substituting in $$\hat{p}=m\hat{v}$$ kinetic energy. We define linear combination as

$$\hat{\alpha}=a\hat{x} \;\mathrm{i} b\hat{p}$$ . Find the real constants <img

src=„https://latex.codecogs.com/gif.latex?a“> a $b$ , such that the Hamiltonian will have the form of

<img src=„https://latex.codecogs.com/gif.latex?\hat{H}=\hbar \omega \left(\hat{\alpha} ^{\dagger}\hat{\alpha}+\frac{1}

{2}\right)\,,“> where $$\hat{\alpha} ^{\dagger}$$ is the complex conjugate <img

src=„https://latex.codecogs.com/gif.latex?\hat{\alpha}“>.

• Show from your knowledge of canoninc commutation relations for

$$\hat{x}$$

and $$\hat{p}$$ that the following is true

<img src=„https://latex.codecogs.com/gif.latex?\left[\hat{\alpha},\hat{\alpha}\right]=0\,,\quad\left[\hat{\alpha} ^{\dagger},\hat{\alpha} ^

{\dagger}\right]=0\,,\quad\left[\hat{\alpha} ,\hat{\alpha} ^{\dagger}\right]=1\,.“>

• In the spectrum of the oscilator there will surely be the state with the lowest possible energy which corresponds to the smallest possible

amount of oscilating. Lets call it $$|0\rangle$$ . This state must fulfill <img

src=„https://latex.codecogs.com/gif.latex?\alpha |0\rangle =0“>. Show that its energy is equal to $$\hbar\omega/2$$ , ie. $$\hat{H}|0\rangle=\hbar\omega/2|0\rangle$$ . Furthermore prove that if $$\alpha |0\rangle \neq 0$$ then we have a contradiction with the fact that <img

src=„https://latex.codecogs.com/gif.latex?|0\rangle“> has a minimal energy ie. <img src=„https://latex.codecogs.com/gif.latex?\hat{H}\alpha |0\rangle=E\alpha

$$E<\hbar\omega/2$$ . All the eigenstates of the Hamiltonian can be described

as $$\left(\alpha^{\dagger}\right) ^n|0\rangle$$

for $$n=0,1,2,\dots$$ Find the energy of these states, in other words find such numbers <img

src=„https://latex.codecogs.com/gif.latex?E_n“> that <img src=„https://latex.codecogs.com/gif.latex?\hat{H}\left(\alpha^{\dagger}\right) ^n|0\rangle=E_n\left

(\alpha^{\dagger}\right)^n|0\rangle“>.

Tip Use the commutation relation for $$\hat{\alpha}^{\dagger}$$ a <img

src=„https://latex.codecogs.com/gif.latex?\hat{\alpha}“>.

• How will the spectrum of an open string on a mass level $M=2⁄α′?$ How many possible states of the string on this level?
• If we consider the interaction of tachyons with other strings, we would find out, že ho můžeme popsat přibližně jako částici pohybující se v nějakém potenciálu. We consider a model of a string that is fastened on a unstable D-brane. The relevant potential of the tachyon is defined by

$$V(\phi)=\frac{1}{3\alpha'}\frac{1}{2\phi _0}(\phi-\phi _0)^2\left (\phi \frac{1}{2}\phi _0\right )\,,$$

where $$\alpha'$$

• The theory of superstrings enables the description of fermions. For their description one needs anticomutating variables. For those one creates an anticomutator instead of a comuator with the relation

$$\{A,B\}=AB BA$$

Find two such $$2\times 2$$