Waveparticle, wave packet, dynamics: Quantum dots

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1. Dengan menggunakan akurasi 1%, tentukan kemungkinan posisi bola tenis dengan berat 0.05 kg yang dilempar dengan kecepatan 10 m/s. (Diketahui nilai h = 6.62607015×10⁻³⁴ J⋅Hz⁻¹, dan ħ = 1.054571817×10⁻³⁴ J⋅s). Dengan cara yang sama, tentukan kemungkinan posisi electron dengan massa 9.11 x 10^-31 kg yang bergerak dengan kecepatan 10 m/s. Jelaskan pendapat anda dengan perbedaan nilai yang didaptkan

2. Elektron dilempar dengan kecepatan 5 m/s dan dia bergerak seperti gelombang. Jika frequency gelombang electron 8×10⁻²⁰ s^-1. Tentukan amplitude dari gelombang electron tersebut! Selanjutnya, dengan akurasi 1%, tentukan kemungkinan besar momentum yang dimiliki jika electron tersebut bertumbukan dengan electron yang lain!

3. Dengan menggunakan akurasi 20%, tentukan kemungkinan posisi bola besi dengan berat 1 kg yang dilempar dengan kecepatan 100 m/s. (Diketahui nilai h = 6.62607015×10⁻³⁴ J⋅Hz⁻¹, danħ = 1.054571817×10⁻³⁴ J⋅s).

4. Formulasikan dan tentukan frekuensi dan amplitude electron yang bergerak dengan kecepatan 10¹⁰ m/s. Gunakan massa electron di literature

5. Hitung kecepatan peluru dan electron yang di tembak pada waktu yang sama. Jika diketahui kemungkinan posisi peluru setelah ditembak sebesar 10^-20 m, sedangkan electron sebeasar 10^-4 cm (gunakan akurasi 1%). Gunakan massa electron yang ada diliteratur, sedangkan mass peluru 5 m/s.

6. Holmium metal ketika di XRD memiliki puncak tertinggi pada 2 theta 30 yang merefleksikan bidang 111. Panjang gelombang X-ray 1.5418 Å. Tentukan uncertainty position dari phonon yang terbentuk dari holmium metal, jika akurasinya 1%!

7. Tentukan uncertainty position electron yang dianggap bergerak melingkar didalam metal Fe.

8. Jelaskan beberapa model dan material untuk quantum circuit. Apa alasannya?

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Wave particle, wave packet, dynamics:

Quantum dots

Tahta Amrillah, PhD

Program Studi Rekayasa Nanoteknologi Fakultas Teknologi Maju dan Multidisiplin

Semester Genap 2021/2022

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Outline:

Introduction

Single electron in quantum dots

Many electron in quantum dots

Dynamics in quantum dots

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Introduction

Applications in solar cells

Silicon → Expensive, complicated fabrication, maximum PCE is around 20%

Thin-film → Expensive, maximum PCE is around 20%

DSSC → Low PCE less than 20%

Perovskite → Unstable, toxic

How to overcome those problems? → Quantum dot integrated solar cells increase the solar cells performance Quantum dot arrays in p–i–n cells (in Silicon and thin film solar cells)

Quantum dots dispersed in organic semiconductor polymer matrices (perovskite and DSSC) Quantum dot-sensitized nanocrystalline TiO₂ solar cells (DSSC)

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Introduction

Applications in Biomedical

Drug delivery Contrast agent Biosensor

Neuromorphic computing

How to realize those ideas? → Quantum dot integrated increase the biocompatibility, sensing, storage, etc.

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Introduction

http://dx.doi.org/10.3184/003685015X14311048821595

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Introduction

• Quantum nanodots

https://doi.org/10.1039/D0RA03938A

•Quantum dot is one of example of 0D materials

•It called quantum nanodot because they are in quantum size regime characterized by the discretization of the energy levels inside the material

•The quantum size regime is obtained when the dimensions are smaller than the exciton Bohr radius (for example in CdS such a threshold value is about 5.4nm)

Int. to Nanoscience and nanotechnology, G. L. Hornyak. CRC Press

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Introduction

Quantum physics will talk about the electron inside nanomaterials; quantum nano dots

We discuss electron inside a simple box

Example:

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Introduction

→ We go back to quantum physics

Some trajectories of a particle in a box according toNewton's laws of classical mechanics(A), and according to the Schrödinger equationof quantum mechanics(B–F). In (B–F), the horizontal axis is

position, and the vertical axis is the real part (blue) and imaginary part (red) of thewavefunction. The states (B,C,D) are energy eigenstates, but (E,F) are not.

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Introduction

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Introduction

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Introduction

→ Now, how about an electron in various low dimensional materials?

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Introduction

→ Now, how about an electron in various low dimensional materials?

Let say:

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Single electron in quantum dots

Quantum states of an electron in quantum dots

Energy levels in a quantum dot

V. V. Mitin, et al. Quantum Mechanics for Nanostructures

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Single electron in quantum dots

Energy levels in a quantum dot

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Many electron in quantum dots

The number of states and density of states in quantum dots

The dependences of (a) the number of states, N(E), and (b) the density of states, g(E), in a rectangular box with potential barriers of infinite height.

How about if not only one electron?

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Many electron in quantum dots

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Double-quantum-dot structures (artificial molecules)

A schematic picture of a double-quantum-dot structure (a) and its potential profile U(x) (b). A and B denote different semiconductor materials, which constitute quantum dots and barriers

between them.

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The wavefunctions of a double-quantum-dot structure, ψs and ψa, when the quantum dots are far from each other.

Double-quantum-dot structures (artificial molecules)

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Dynamics in quantum dots

wavefunctions ψs and ψa, when the width of the barrier between the two quantum dots is equal to zero.

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The dependences of the energies of the

symmetric, E(1) x , and antisymmetric, E(2) x , states of the doublequantum-dot structure on the width of the barrier, d, separating

individual quantum dots.

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Dynamics in quantum dots

The wavefunctions of a double-quantum-dot structure, ψs and ψa, when the quantum dots are far from each other.

We go back to double quantum dot with a distance

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Dynamics in quantum dots

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The wavefunctions of a double-quantum-dot

structure, ψs and ψa, when the quantum dots are close to each other

Then, the double quantum dot with a distance move to close each other!

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Dynamics in quantum dots

The wavefunctions of a double-quantum-dot

structure, ψs and ψa, when the quantum dots are close to each other

Then, the double quantum dot with a distance move to close each other!

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Dynamics in quantum dots

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What is the purpose of our calculation? ➔ remember core-shell structure for any applications, e. g. drug delivery!

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What is the purpose of our calculation? ➔ remember core-shell structure for any applications, e. g. drug delivery!

To use QDs for biological applications, one must passivate the core QD with a thin layer of non-toxic high band gap material such as ZnS or ZnSe⁵⁵. Such core/shell QDs have many advantages over unpassivated ones. For example, the chemical and optical stability of QDs can be maintained for longer

periods of time when they are passivated with higher band-gap

semiconductor materials⁵⁶. The shell also significantly reduces the toxicity of the QDs, which makes them more suitable for biological applications. To

date, many types of core/shell QDs were fabricated. However, only CdSe/ZnS, CdTe/ZnS, and CdSe/CdS/ZnS have been commonly used forin vitro andin vivoimaging⁵⁷. Such core/shell QDs are created by epitaxially growing a higher band-gap semiconductor material around the core.

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Wave particle, wave packet, dynamics:

Nano heterostucture

Tahta Amrillah, PhD

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Outline:

Introduction

Quantum wires

Quantum wells

Dynamics in Nano heterostucture

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Introduction

What is nano heterostructures?

• 0D-0D core shells

• 0D-1D nanocomposites

• 2D-2D multilayer

• 3D-3D nanocomposites, core shells

Examples:

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Introduction

rsos.royalsocietypublishing.org, R. Soc. open sci.5: 1803876

(35)

Introduction

Possible methods of circumventing the 31% efficiency limit for

thermalized carriers in a single–band gap absorption threshold solar quantum conversion system. (A) Intermediate-band solar cell; (B) quantum-well solar cell.

DOI: 10.1126/science.1137014

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Quantum Wires

Different types of nanostructures: (a) a quantum well, (b) a quantum wire, and (c) a quantum dot.

A and B denote the materials which constitute the nanostructures, with A being the material of the nanostructure itself and B being the material of the surrounding matrix. The nanostructure is referred to as free-standing if the surrounding matrix, B, is vacuum

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Quantum Wires

In general: In quantum wire:

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Quantum Wires

Quantum wire: the quantized energy levels, E11 and E12, as well as the dependences of the electron

energy E_nxny on the momentum p_z

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Quantum Wires

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Quantum Wires

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Quantum wells

In quantum wire:

In quantum wells:

In general:

Quantum well: quantized energy levels, E1 and E2, and the dependences of the electron energy, E_nx , on the momentum

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Quantum wells

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Quantum wells

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Quantum wells

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Quantum wells

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Lets we consider the simplest combination of similar low dimensional materials in superlattice or periodic structure

The potential profile of the superlattice in the direction of the periodicity (the x-direction). U₀is the height of the barrier which separates neighboring quantum dots, d is the width of the potential barrier, and L_xis the width of the potential well. A denotes the material of the quantum dot and B denotes the material of the barriers that separate quantum dots. The first energy miniband is labeled 1 and the second is labeled 2

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Dynamics in Nano heterostucture

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Dynamics in Nano heterostucture

A superlattice composed from quantum wires. A denotes the material of the quantum wires and B the material of the

barriers which separate the quantum wires

A superlattice composed from quantum wells. A denotes the material of the quantum wells and B the material of the barriers, which separate the quantum wells.

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Heisenberg Operator Approach

(Introduction)

Tahta Amrillah, PhD

(53)

Outline:

Introduction

Heisenberg Uncertainty

Heisenberg Solving in Nanomaterial

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Introduction

“Big Picture” of Industry Automation

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Introduction

Mechanics in the production process

We can calculate the movement!

Force, velocity, distance, etc. →

Physicsunderstanding is required

How about in microchip in our computers?

What is moving, electron?

Can calculate the movement!

Force, velocity, distance, etc. → Modern and quantum Physics

understanding is required

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Introduction

https://www.youtube.com/watch?v=2z9qme_ygRI https://www.youtube.com/watch?v=rkbjHNEKcRw

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Introduction

Electrons exhibit particle–wave duality. The de Broglie equation relates the momentum of an

electron (a particle phenomenon) to wavelength

https://sites.google.com/site/atomictheorytimeline707/privacy-policy

Int. to Nanoscience and nanotechnology, G. L. Hornyak. CRC Press

Back to probability of electron

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Heisenberg Uncertainty

However, electron cannot move like basket balls

m

v s t

m

“In macroscopic view, everything is certainty – deterministic!”

v s t

How about we changes the basket ball with electron!!!

Can we observed it?

Can we calculate the velocity?

Can we observed where the electron move after collision?

How much the momentum?

etc.

Clue: Probability

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Heisenberg Uncertainty

• In the world of very small particles, one cannot measure any property of a particle without interacting with it in some way

• This introduces an unavoidable uncertainty into the result

• One can never measure all the properties exactly

Werner Heisenberg (1901-1976)

Image in the Public Domain

Heisenberg realized that ...

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• Shine light on electron and detect reflected light using a microscope

• Minimum uncertainty in position

is given by the wavelength of the light

• So to determine the position

accurately, it is necessary to use light with a short wavelength

BEFORE ELECTRON-PHOTON

COLLISION

electron incident

photon

Measuring Position and Momentum of an Electron

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• By Planck ’ s law E = hc/λ, a photon with a short wavelength has a large energy

• Thus, it would impart a large ‘ kick ’ to the electron

• But to determine its momentum

accurately, electron must only be given a small kick

• This means using light of long wavelength !

AFTER

ELECTRON-PHOTON COLLISION

recoiling electron scattered

photon

Measuring Position and Momentum of an Electron

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• It is impossible to know both the position and momentum exactly, i.e., Δx=0 and Δp=0

• These uncertainties are inherent in the physical world and have nothing to do with the skill of the observer

• Because h is so small, these uncertainties are not observable in normal everyday situations

Implications

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• A pitcher throws a 0.1-kg baseball at 40 m/s

• So momentum is 0.1 x 40 = 4 kg m/s

• Suppose the momentum is measured to an accuracy of 1 percent , i.e.,

Δp = 0.01p = 4 x 10^-2 kg m/s

• The uncertainty in position is then

• No wonder one does not observe the effects of the uncertainty principle in everyday life!

Example of Baseball

• Same situation, but baseball replaced by an electron which has mass 9.11 x 10^-31 kg traveling at 40 m/s

• So momentum = 3.6 x 10^-29 kg m/s

and its uncertainty = 3.6 x 10^-31 kg m/s

• The uncertainty in position is then Example of Electron

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Compute the wavelength of an electron (m = 9.1x10^-31 [kg]) moving at 1x10⁷ [m/s].

λ = h/mv

= 6.6x10^-34 [J s]/(9.1x10^-31 [kg])(1x10⁷ [m/s])

= 7.3x10^-11 [m].

= 0.073 [nm]

Heisenberg Uncertainty

• This is immeasureably small (impossible). For ordinary “everyday objects,”

we don’t experience that. But for sure MATTER CAN BEHAVE AS A WAVE

Once more, example of Bullet

• Compute the wavelength of a 10 [g] bullet moving at 1000 [m/s].

λ = h/mv = 6.6x10^-34 [J s] / (0.01 [kg])(1000 [m/s])

= 6.6x10^-35 [m]

But, what about small particles ?

These electrons

have a wavelength in the region of X-rays

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Heisenberg Uncertainty

• The observer is objective and passive

• Physical events happen independently of whether there is an observer or not

• This is known as objective reality

Classical World

• The observer is not objective and passive

• The act of observation changes the physical system irrevocably

• This is known as subjective reality

Role of an Observer in Quantum Mechanics

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Both uncertainty!

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or

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Note: Heisenberg only proved relation (devided by 2) for the special case of Gaussian states

The original

Expanded:

Dont be confuse!

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Heisenberg Solving in Nanomaterial

Movement of some small nanoparticles (or quantum dots → electron)

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https://doi.org/10.1039/C8DT01631C Covalently linking photosensitizers and catalysts in an inorganic–organic hybrid

photocatalytic system is beneficial for efficient electron transfer between these components.

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Heisenberg Solving in Nanomaterial

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In quantum physics

Basically the Schrodinger picture time evolves the probability distribution, the Heisenberg picture time evolves the dynamical variables and the interaction picture time evolves a little bit of both

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Example in very simple case: 1D harmonic oscillator

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Heisenberg Solving in Nanomaterial

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Heisenberg exchange interaction, J_ij Example of applications Heisenberg picture

In Article: Macrospin model of an assembly of magnetically coupled core-shell nanoparticles

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Heisenberg Operator Approach : Nano

Mechanical Oscillator

Tahta Amrillah, PhD

(84)

Outline:

Introduction

Nanomechanical Oscillator

Real Apllications

(85)

Introduction

Bim-salabim jadi solar cells Why physics???

(86)

Introduction

• Massa yang dihubungkan pada pegas (Gambar disamping) .

• Pegas ditarik ke kanan menggunakan gaya 𝐹

_𝑇

,

• Pegas memberi reaksi kekiri sebesar 𝐹

_𝑠

(Hukum aksi-reaksi 𝐹

_𝑇

= − 𝐹

_𝑠

). Gaya pegas 𝐹

_𝑠

= 𝑚𝑎

_𝑥

• Jika gaya tarik tidak melampui batas elastistas pegas, pertambahan panjang pertambahan panjang pegas 𝑥 sebanding dengan gaya tariknya 𝐹

_𝑇

= 𝑘𝑥

• Dengan demikian, berlaku 𝐹

_𝑠

= −𝑘𝑥 = 𝑚𝑎

_𝑥

2

0

2

0

x

d x k

ma kx x

dt m

 + =  + =

Oscillator in classical physics!

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Introduction

• Persamaan gerak pada pegas 𝑑

²

𝑥

𝑑𝑡

²

+ 𝑘

𝑚 𝑥 = 0

• 𝑥 𝑡 = 𝐴 cos 𝜔𝑡 + 𝜙 , maka:

• Dengan mensubtitusikan persamaan-persaan ini

• Pada persamaan gerak pegas, akan didapatkan:

( )

2 2

cos 0

k A t

m    

 −  + =

 

 

2 k

 = m

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Introduction

• Misalkan, Simpangan osilasi 𝑥 𝑡 = 𝐴 cos 𝜔𝑡 + 𝜙

• Kecepatan

• Percepatan

( ) dx t ( ) sin ( )

v t A t

dt   

= = − +

( ) ( )

2

( ) d x t2 cos

a t A t

dt   

= = − +

Simpangan Kecepatan Percepatan

+ +/- -

- -/+ +

Maksimum Nol Maksimum

Nol Maksimum nol

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( )

2 2 2 2

2 2 2

0.5 0.5 sin

0.5 0.5 cos

0.5 0.5

EK mv m A t

EP kx m A t

E EK EP m A kA

  



= = +

= + = =

(90)

Introduction

Oscillator in Quantum mechanics!

rsos.royalsocietypublishing.org, R. Soc. open sci.5: 1803876

(91)

Introduction

Possible methods of circumventing the 31% efficiency limit for

thermalized carriers in a single–band gap absorption threshold solar quantum conversion system. (A) Intermediate-band solar cell; (B) quantum-well solar cell.

DOI: 10.1126/science.1137014

(92)

Introduction

Potential energy:

Displacement due to potential energy:

With:

Frequency:

Amplitude:

Newton's lawsofclassical mechanics(A), and according to theSchrödinger

equationofquantum mechanics(B–F). In (B–F), the horizontal axis is position, and the vertical axis is the real part (blue) and imaginary part (red) of thewavefunction. The states (B,C,D) areenergy eigenstates, but (E,F) are not.

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Introduction

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A one-dimensional harmonic oscillator

Wave function

where the quantum number, n, has the values n = 0, 1, 2,... The quantum number n defines the energy of oscillatory motion of a quantum oscillator and it is called the oscillatory quantum number.

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Nanomechanical Oscillator

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Some case; Phonon:

In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids

A phonon is the quantum mechanical description of an elementary vibrational motion in which a lattice of atoms or molecules uniformly oscillates at a single frequency

normal modes of phonon are wave-like phenomena in classical mechanics, phonons have particle- like properties too, in a way related to the wave–particle duality of quantum mechanics

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Nanomechanical Oscillator

Some case; Phonon:

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Nanomechanical Oscillator

Velocity group:

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Is there only phonon and photon?

There are a lot of thing/phenomenon we can observe using quantum mechanics!!!

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Real Applications

Also, nanomechanical oscillator could be used in some precision

measurement tools!

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Heisenberg Operator Approach: Quantum LC

Circuit

Tahta Amrillah, PhD

(104)

Outline:

Introduction

Classical LC Circuit

Quantum LC Circuit

(105)

Introduction

Konduktor mempunyai Elektron bebas, yang

• Sering mengalami hamburan dari kisi kristal (ion positif) terjadi gerak random, tetapi tidak menghasilkan arus

• Pemberian medan listrik menghasilkan kecepatan drift yang kecil.

• Kecepatan drift ini menghasilkan gerak muatan neto - yaitu arus listrik, I → Satuan Amp (Ampère), 1 A = 1 C s

^–1

Elektron di Dalam Konduktor

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Introduction

Arus Listrik

R ε, r

- + I

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Introduction

Suatu resistor atau tahanan mempunyai sifat resistansi, satuan Ohm (Ω) dapat dinyatakan dalam hubungan sebagai :

 = resistivitas, dengan satuan  m l = panjang kawat, meter (m)

A = luas kawat, m

²

Besarnya Arus yang mengalir di dalam komponen listrik tergantung pada beda potensial ujung-ujungnya, V.

(hukum Ohm)

Satuan resistansi (tahanan) : ohm,  (1  = 1 volt per amp)

Resistansi Listrik

A R  l

=

R

I = V

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Introduction

Pengurangan energi dari muatan Q yang berpindah pada beda potensial V adalah Q V.

Daya yang hilang (rate kehilangan energi) adalah P = (dQ/dt )V = I V.

Dengan V = I R, maka daya adalah :

Ini adalah daya listrik yang oleh tahanan dirubah menjadi panas, dan panas yang timbul persatuan waktu adalah : dH/dt = I

²

R

Daya Listrik

R I

a b

2

V

2

P IV I R

= = = R

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Introduction

RANGKAIAN ARUS SEARAH

Sumber tegangan atau disebut juga emf (electro motive force) didefinisikan sebagai energi persatuan muatan :

ε = dW/dt Volt atau joule/coulomb

Perhatikan gambar dibawah ini, sebuah emf mempunyai tahanan dalam, r, dihubungkan dengan tahanan luar R.

ε I = I

²

R + I

²

r

I = ε/(R+r)

^R

ε, r

- + I

(110)

Introduction

Bila kemudian dipasang motor : ε I = I

²

R + I

²

r + I

²

r’ + ε’I

ε - ε’ = I (R + r + r’) Σε = Σ I r

_n

Hk Kirchoff II :

Jumlah aljabar sumber tegangan (emf) = jumlah aljabar dari perkalian arus dan tahanan pada suatu loop (rangkaian tertutup)

I = (ε - ε’ )/(R + r + r

’) I = Σε /Σ r

_n

ε positif bila searah dengan I, dan R selalu positif Σ I = 0 Hk Kirchoff I :

jumlah aljabar dari arus pada suatu titik cabang, a, dalam suatu rangkaian listrik = 0

R₁ R₂ R₃

I₁ a I₂ I₃

I masuk positif dan I keluar negatif

R

ε, r -

+

I

- +

ε’, r’

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Introduction RANGKAIAN ARUS BOLAK BALIK

S

_R

v_S

Resistansi (R, satuan Ohm) dihubungkan dengan Tegangan (emf) dengan fungsi : v

_S

= V

_m

sin 2ft = V

_m

sin ωt (volt)

v

_S

= tegangan sumber fungsi waktu (volt) V

_m

= tegangan maksimum (volt)

f = frekuensi (di Indonesia 50 Hz) Menurut hukum ohm

V =I R

V

_m

sin ωt = i R → i = (V

_m

/R) sin ωt = i

_m

sin ωt dengan i

_m

= V

_m

/R = arus maksimum

Tegangan dan arus berfase sama (sefase)

i

i_m V_m

y

t

i_m V_m y

x

i_m V_m

t y

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Introduction

t cos

I

i

_L

=

_m



Reaktansi induktif, X

_L

mengalir melewati induktor L, tegangannya diperoleh dari :

   

) 90 t

( cos LI

)}

90 t

{- cos LI

t) (- sin LI

v

t sin I

- L t

cos dt I

L d dt

L di

o m

m L

m m

L

+



=

−



=



=





=



=

L

= v

ωLI

_m

= V

_m

tegangan maksimum

Reaktansi induktif X

_L

= L dalam satuan ohm ().

V

_m

= I

_m

X

_L

Bila arus :

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Introduction

Kesimpulan :

tegangan pada induktor L

mendahului 90

⁰

dari arus yang melewatinya.

v

_L

I

θ

v

_L

I

90⁰

ωt v, i

i_L (t) v_L (t)

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Introduction

   

( )

( t 90 )

cos

i

90 t

cos C

V C

t sin

- C t

cos dt V

C d dt

C dv

0 C

0 m

m C

+



=



−



−



=



−



=



=



=

m

m C

CV

V )

t (

sin

V i

Reaktansi Kapasitif, X

_C

melintang pada ujung kapasitor C, arusnya adalah:

t cos

V

v

_C

=

_m



Bila tegangan :

dt C dv dt

dQ =

_C

C

= i

ωCV

_m

= I

_m

arus maksimum

Reaktansi capasitif X

_C

= 1/C dalam satuan ohm ().

V

_m

= I

_m

X

_C

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Introduction

Kesimpulan :

Arus yang melewati kapasitor C mendahului 90

⁰

dari tegangannya.

v, i

i_C(t) v_C (t)

90⁰

ωt

θ

v_L I_L

C

I

_L

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Introduction

Resistansi (R, satuan Ohm), induktansi (L, satuan Henry) dan capasitor (C, satuan Farad) dihubungkan dengan Tegangan (emf) dengan fungsi :

RANGKAIAN ARUS BOLAK BALIK

v

_S

= V

_m

sin 2ft = V

_m

sin ωt (volt)

S

R L C v_S

I

_m

= V

_m

/Z Ampere (A)

Diperkenalkan impedansi :

Z = (R

²

+ X

²

)

^1/2

satuan ohm ( Ω )

I

_S

tidak dapat dituliskan

sebagai I

_s

= v

_s

/Z , karena I

_s

dan v

_s

ada

beda fase,yaitu : φ =tan

^-1

(X/R) (derajat)

v

_S

= tagangan sumber fungsi waktu (volt) V

_m

= tegangan maksimum (volt)

f = frekuensi (di Indonesia 50 Hz)

R = resistansi, Ω X = reaktansi, Ω

X

_L

= (2f L) = ωL, Ω = reaktansi induktif

X

_C

=1/(2fC) = 1/(ωC), Ω = reaktansi capasitif

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Introduction

S

R L C v_S

RANGKAIAN ARUS BOLAK BALIK

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Introduction

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Energy Kinetic

Lagrangian = analisis di dalam mekanika yang tidak

mempertimbangkan keberadaan gaya dalam pergerakan yang timbul

Penyelesaian umum:

The Euler–Lagrange equation Jadi, the electrical charge Q sebesar:

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Classical LC Circuit

Jadi, the electrical charge Q sebesar:

angular frequency of oscillations

Continue with Hamiltonian (Hamiltonian suatu sistem adalah operator yang sesuai dengan energi total sistem tersebut, termasuk energi kinetik dan energi potensial)

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Classical quantum mechanics for LC circuits

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Quantum LC Circuit

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Quantum LC Circuit

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Quantum LC Circuit

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Quantum LC Circuit

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Quantum LC Circuit

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Quantum LC Circuit

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Quantum LC Circuit

Some applications

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Quantum LC Circuit

SQUID magnetometer

Very sensitive magnetic detection in some nanomaterials

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Quantum LC Circuit

Single electron transistor

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Quantum LC Circuit

I hope you the one who build this technology in Indonesia!

Next generation devices

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