i  ∂∂ t = − 2  m 2 ∂ ∂ 2 x 2 + U

(1)

Kvantfysik

[1]

Uppdaterad: 171211

Har jag använt någon bild som jag inte får använda? Låt mig veta så tar jag bort den.

christian.karlsson@ckfysik.se

[1] Läget runt 1920

[2] Materiens vågegenskaper (de Broglie)

[3] Men kan det vara så här? (Elektroner och gitter)

[4] Men kan det vara så här? (Elektroner och dubbelspalt) [5] Materiens vågegenskaper (Schrödinger – Born)

[6] Kvantmekanik (tidig) [7] Kvantmekanik (tidig)

[8] Heisenbergs obestämdhetsrelation [9] Läget runt 1927

[10] Efter 1927 [11] Läget idag

[12] Läget idag i  ∂ ψ

∂t = − 

²

2m

∂

²

ψ

∂x

²

+U ψ

[2]

(2)

Kvantfysik

[3]

[4]

X

Richard Feynman

Märklig och ointuitiv – men inte ologisk!

Do not keep saying to yourself, if you can possibly avoid it,

”But how can it be like that?” because you will get ”down the drain”, into a blind valley from which nobody has yet escaped.

Nobody knows how it can be like that.

Niels Bohr

But, but, but ... if anybody says he can think about quantum theory without getting giddy it merely shows that he hasn’t understood the first things about it!

[5]

[6]

FSS

(3)

Några milstolpar i fysiken

Newton (Principia)

Young (Dubbelspaltexpt.) Maxwell (E och M)

Thomson (elektronen) Planck (energikvanta) Einstein (ljusenergi

kvantiserad) Bohr (ny atommodell)

1687

~1803

~1864 1897 1900 1905

1913 Spec. rel.teori

[7]

X

(4)

Några milstolpar i fysiken

Newton (Principia)

Young (Dubbelspaltexpt.) Maxwell (E och M)

Thomson (elektronen) Planck (energikvanta) Einstein (ljusenergi

kvantiserad) Bohr (ny atommodell)

1687

~1803

~1864 1897 1900 1905

1913 Spec. rel.teori

[7]

X

(5)

Läget runt 1920

X

[8]

FSS

(6)

Läget runt 1920

1 Partikelegenskaper Vågegenskaper

Materia

Strålning

Ja! ^(mekanik)

p = mv

W

_k

= mv

²

2

p =

γ

^mv ^E_k = (

γ

−1)mc² R = p

Ja! (fotonmodellen)

?

light!waves and particles" are mutually exclusive because they appear independently in different experiments. To elimi- nate this conceptual difficulty we have designed an apparatus in which the particle and wave aspects can first be demon- strated individually. Then the same apparatus is used to vi- sualize the real-time evolution of individual quantum events to a classical wave pattern. The use of the same light source and the same interferometer is important to convince stu- dents that we can investigate the two aspects of light with the same apparatus.

Two-beam interference phenomena are often explained on the basis of Young’s double slit experiment by displaying the well known interference pattern on a distant screen. Al- though this example is well suited for a theoretical discussion and most easily realized using a laser pointer and a double slit, it is not practical for advanced demonstration experiments because it does not allow the variation of system parameters in a simple way. In the present experiment we have chosen a Mach-Zehnder interferometer in which a large spatial separation of the two interfering beams can be easily realized, permitting several manipulations, such as the ad- justment of the path length difference and the relative angle of the interfering beams, and, most importantly, the easy blocking of one of the two beams. The macroscopic dimen- sions of the Mach-Zehnder interferometer allow the observer to see all components from the light source via the genera- tion and recombination of the interfering beams up to their detection.

A green laser pointer was chosen as the light source because it has a sufficiently long coherence length for the easy alignment of the interferometer. The intensity of the green beam and its wavelength near the vicinity of the eye’s maxi- mum sensitivity ensure that even expanded interference patterns are easily visible in a large auditorium.

Our main design criterion was to have the apparatus as simple and pedagogical as possible while also offering the flexibility to vary certain parameters to illustrate several as-

pects of the phenomena. The equipment is designed for dem- onstrations in a large auditorium. The interferometer is mounted on an aluminum plate tilted by 45°, so that all components can be easily seen. If necessary, a webcam can be used to project a close-up of the interferometer table. As mentioned, the expanded fringe pattern using the full laser intensity can easily be seen from a distance without addi- tional tools. Individual photon events can be seen as pulses on an oscilloscope, or heard as clicks using audio equipment.

All relevant electronic signals !photomultiplier pulses and photodiode signals" can easily be projected using equipment such as a digital oscilloscope equipped with a video port or a USB-based oscilloscope. Attention was paid to obtain stable pictures and good visibility of all the components and projected signals. Last but not least, we have made an effort to reduce component cost as much as possible, and to give the apparatus a pleasant look.

III. EXPERIMENTAL SETUP

The scheme of the experimental apparatus is shown in Fig.

2 and a photo of its main components in Fig.3. The light source is a 5 mW green!!=537 nm" laser pointer. The bat- teries in the laser pointer were replaced by electrical contacts so that the laser could be driven by an external power supply.

We found that the spectral width of the laser radiation and hence its coherence length depends on the operating voltage and a randomly chosen pointer has its own optimal voltage for the highest fringe contrast. Once set correctly and after a warm-up time of several minutes this voltage gives repro- ducible results on daily basis.

The standard Mach-Zehnder interferometer has two beam splitters and two mirrors !all 1 in. optics" arranged in a 18

"18 cm²square !see Figs.2and3". One of the mirrors is mounted on a low-voltage piezotransducer that allows the voltage-controlled variation of the path length difference

!#5 V per fringe".

Fig. 1.!Color online" From particles to waves: Detection of light diffracted from a double slit on a photon by photon basis using a single-photon imaging CCD camera. Although single frames show an apparently random distribution of photon impact points, their integration reveals the classical fringe pattern.

138 Am. J. Phys., Vol. 76, No. 2, February 2008 T. L. Dimitrova and A. Weis 138

light !waves and particles" are mutually exclusive because they appear independently in different experiments. To elimi- nate this conceptual difficulty we have designed an apparatus in which the particle and wave aspects can first be demon- strated individually. Then the same apparatus is used to vi- sualize the real-time evolution of individual quantum events to a classical wave pattern. The use of the same light source and the same interferometer is important to convince stu- dents that we can investigate the two aspects of light with the same apparatus.

The standard Mach-Zehnder interferometer has two beam splitters and two mirrors!all 1 in. optics" arranged in a 18

"18 cm² square!see Figs.2 and3". One of the mirrors is mounted on a low-voltage piezotransducer that allows the voltage-controlled variation of the path length difference

!#5 V per fringe".

[9]

Ja! (elektromagnetism, optik)

W = hf

p = h

λ

^{E, B}

Dubbelspaltexperiment med fotoner (2005)

FSS

(7)

Läget runt 1920

X

2 and a photo of its main components in Fig. 3. The light source is a 5 mW green !!=537 nm" laser pointer. The bat- teries in the laser pointer were replaced by electrical contacts so that the laser could be driven by an external power supply.

"18 cm² square !see Figs. 2 and 3". One of the mirrors is mounted on a low-voltage piezotransducer that allows the voltage-controlled variation of the path length difference

!#5 V per fringe".

Fig. 1. !Color online" From particles to waves: Detection of light diffracted from a double slit on a photon by photon basis using a single-photon imaging CCD camera. Although single frames show an apparently random distribution of photon impact points, their integration reveals the classical fringe pattern.

[9]

FSS

(8)

Materiens vågegenskaper

[10]

2 de Broglie (1924):

Varje partikel kan tillskrivas en våglängd λ = h p

Plancks konstant

partikelns rörelsemängd

“materievåglängd”

[11]

p = h

λ ⇒ λ = h Fotoner: p

Einstein (1916):

(9)

Materiens vågegenskaper

[10]

Ger en slags förklaring till

väteatomens kvantiserade energinivåer:

2 de Broglie (1924):

Varje partikel kan tillskrivas en våglängd λ = h p

Plancks konstant

“materievåglängd”

[11]

villkoret

2 π r = n λ

måste vara uppfyllt

p = h

Einstein (1916):

Jfr stående våg på sträng:

(10)

Materiens vågegenskaper

[10]

Ger en slags förklaring till

väteatomens kvantiserade energinivåer:

2 de Broglie (1924):

Varje partikel kan tillskrivas en våglängd λ = h p

Plancks konstant

“materievåglängd”

Bekräftades 1927

G. P. Thomson (UK)

Davisson & Germer (USA)

[11]

[13]

[12]

villkoret

2 π r = n λ

måste vara uppfyllt

p = h

Einstein (1916):

Jfr stående våg på sträng:

(11)

Men kan det vara så här?

3 FSS

(12)

Men kan det vara så här?

Elektronkanon

3

U_acc

+ –

Glödtråd e–

FSS

(13)

Men kan det vara så här?

Vitt ämne som fluorescerar i grönt

när det träffas av elektroner.

Elektronkanon

3

U_acc

+ –

Glödtråd e–

FSS

(14)

Men kan det vara så här?

Tunt lager av grafit. Fungerar som ett gitter*

för elektronerna (p.g.a. den regelbundna atomstrukturen).

Vitt ämne som fluorescerar i grönt

när det träffas av elektroner.

Elektronkanon

3

U_acc

+ –

Glödtråd e–

FSS

(15)

Men kan det vara så här?

Tunt lager av grafit. Fungerar som ett gitter*

för elektronerna (p.g.a. den regelbundna atomstrukturen).

Vitt ämne som fluorescerar i grönt

när det träffas av elektroner.

Elektronkanon

3

U_acc

+ –

Glödtråd e–

Ringar – partiklar har vågegenskaper!

FSS

[14]

(16)

[16]

[15]

4 Men kan det vara så här?

Dubbelspaltmönster med elektroner har observerats!

[17]

https://www.youtube.com/watch?v=5oQWtcfZN4M

FSS

(17)

Materiens vågegenskaper

5 Elektronen – våg eller partikel?

https://www.youtube.com/watch?v=mhfdwH5Kdbc

Jämför med ljusinterferens:

sannolikheten att en elektron träffar ett visst ställe är fördelad som

intensitetsfördelningen från en våg som passerat dubbelspalten

e^–-kanon

Dubbelspalt Skärm

Intensitet

Läge

(18)

Materiens vågegenskaper

5 Elektronen – våg eller partikel?

https://www.youtube.com/watch?v=eCFTVdExxPA

Elektroner – sänds ut och detekteras som partiklar – sannolikheten att hitta en elektron någonstans beskrivs med en våg

Jämför med ljusinterferens:

sannolikheten att en elektron träffar ett visst ställe är fördelad som

intensitetsfördelningen från en våg som passerat dubbelspalten

e^–-kanon

Dubbelspalt Skärm

Intensitet

Läge

Ingetdera!

(19)

Kvantmekanik (tidig)

http://www.youtube.com/watch?v=8GZdZUouzBY

Matematik ¨ar vackert – tre exempel

Arean A under kurvan y = x² mellan x = 0 och 1:

A =

⇧1

0

x²dx =

⇤x³ 3

⌅1 0

= 1

3 (1)

1 0,5

0,5 1

y

A

y = x²

x

Samband mellan e = 2.71828 . . ., i som ¨ar s˚adant att i² = 1 och ⇥ = 3.14159 . . .:

eⁱ = 1 (2)

V˚agfunktionen för en elektron i första skalet i en väteatom (f˚as om man löser Schrödingerekvationen för väteatomen):

⇤₁₀₀(r) = 1

⇥⇥ 1 a0

⇥³₂

e ^a0^r , (3)

d¨ar

a0 = 4⇥ ₀ ²

µe² . (4)

Sannolikheten att hitta elektronen mellan avst˚anden R2 och R1

(fr˚an k¨arnan):

P (R₂, R₁) = 4⇥

R2

⇧

R₁

r²⇤₁₀₀² dr (5)

6 I kvantmekaniken (1925-26) beskrivs partiklar med vågfunktioner ψ( x,t)

har att göra med sannolikheten att hitta en partikel i (x, t)

Schrödinger

0 0,05 0,1 0,15

0 1 2 3 4 5

y =x²e^{−2 x}

[22]

Heisenberg

Born Dirac

(en formulering av)

[21]

[20]

[18] [19]

FSS

(20)

Solvay-konferensen 1927

X

[23]

FSS

(21)

X

Solvay-konferensen 1927

[24]

FSS

(22)

X

Solvay-konferensen 1927

42 år 25 år

44 år 35 år

48 år

27 år 40 år

69 år

35 år

[24]

FSS

(23)

Kvantmekanik (tidig) vs. klassisk mekanik

7 Klassisk mekanik

Kvantmekanik

x(t)

ψ(x,t)

R = m d

²

x dt

²

i  ∂ψ

∂t = − 

²

2m

∂

²

ψ

∂x

²

+U ψ

Lägesfunktion

Vågfunktion

Sannolikheten att hitta partikeln mellan x

₁

och x

₂

:

P(x

₁

, x

₂

) = |ψ |

²

dx

x₁ x₂

∫

Newton II

Schrödingerekvationen

Representerar omgivningens inverkan

1

0 2 3 4 5 m

x

m R

FSS

(24)

X

Kvantmekanik (tidig)

[25]

FSS

(25)

Inte bara elektroner ...

X

Neutroner, He-atomer, C

₆₀

-molekyler och...

FSS

(26)

Inte bara elektroner ...

[27]

X

[26]

... ftalocyanin:

Neutroner, He-atomer, C

₆₀

-molekyler och...

(STM-bilder)

FSS

(27)

Inte bara elektroner ...

[27]

Real-time single-molecule imaging of quantum interference

Thomas Juffmann¹, Adriana Milic¹, Michael Mu¨llneritsch¹, Peter Asenbaum¹, Alexander Tsukernik², Jens Tu¨xen³, Marcel Mayor^3,4, Ori Cheshnovsky^2,5and Markus Arndt¹*

The observation of interference patterns in double-slit experiments with massive particles is generally regarded as the ulti- mate demonstration of the quantum nature of these objects.

Such matter–wave interference has been observed for electrons¹, neutrons², atoms^3,4and molecules^5–7and, in contrast to classical physics, quantum interference can be observed when single particles arrive at the detector one by one. The build-up of such patterns in experiments with electrons has been described as the “most beautiful experiment in physics”^8–11. Here, we show how a combination of nanofabrication and nano-imaging allows us to record the full two-dimensional build-up of quantum interference patterns in real time for phthalocyanine molecules and for derivatives of phthalocyanine molecules, which have masses of 514AMU and 1,298AMU respectively. A laser-controlled micro-evaporation source was used to produce a beam of molecules with the required intensity and coherence, and the gratings were machined in 10-nm-thick silicon nitride membranes to reduce the effect of van der Waals forces. Wide-field fluorescence microscopy detected the position of each molecule with an accuracy of 10 nm and revealed the build-up of a determi- nistic ensemble interference pattern from single molecules that arrived stochastically at the detector. In addition to providing this particularly clear demonstration of wave–particle duality, our approach could also be used to study larger molecules and explore the boundary between quantum and classical physics.

When Richard Feynman described the double-slit experiment with electrons as being ‘at the heart of quantum physics’¹²he was emphasizing how the fundamentally non-classical nature of the superposition principle allows the quantum wavefunction associ- ated with a massive object to be widely delocalized, while the object itself is always observed as a well-localized particle. Recent experiments have focused this discussion by demonstrating the stochastic build-up of interference patterns^11,13, by implementing double-slit diffraction in the time domain^14,15(including down to the attosecond level¹⁶), and by identifying a single molecule as the smallest double-slit for electron interference^17,18that enables funda- mental studies of decoherence¹⁹. The extension of this work²⁰to large molecules requires a sufficiently intense and coherent beam of slow and neutral molecules, a nanoscale diffraction grating, and a detector that offers a spatial accuracy of a few nanometres and a molecule-specific detection efficiency of close to 100%. We achieve that in this work with a combination of micro-evaporation, nanofabrication and nano-imaging.

Our experimental set-up comprises three sections: beam prep- aration, coherent manipulation and detection (Fig. 1). The molecules

need to be prepared such that each one interferes with itself, and all lead to similar interference patterns on the screen. Because the trans- verse and longitudinal coherence functions are determined by the Fourier transforms of the source spatial extension and velocity distribution²¹, we require a good collimation and velocity selection. Under

‘far-field’ conditions we can approximate the molecular wave- functions as plane waves, and the angle u_nof the nth order diffraction peak is given by the equation sin u_n¼ nL/d, where L ¼ h/mv is the de Broglie wavelength, h is Planck’s constant, m is the particle mass, v is the velocity, and d is the period of the diffraction grating.

Massive particles therefore need to be slow to achieve sizable diffraction angles. Although deceleration techniques have been advanced for molecules even as complex as benzonitrile²², effusive beams (Fig. 1b) are still well suited for preparing slow beams of particles a hundred times more massive than that^23,24.

For thermolabile organic molecules, which may decompose when heated to their evaporation temperature, we use a laser micro-source (Fig. 1a), which reduces the heat load to a minimum. A blue diode laser is focused onto a thin layer of molecules deposited on the inside of the entrance vacuum window W₁, which can be moved by a motorized translation stage. Although high temperatures can be reached locally, this affects only the particles within the focus area. In comparison to a Knudsen cell, the heat load to the sample is therefore reduced by two to three orders of magnitude (to several 10 mW). Spectral coherence is achieved by sorting the arriving molecules according to their longitudinal velocity and their respective freefall height in the Earth’s gravitational field²⁵.

The collimation slit S defines the spatial coherence of the molecular beam. The slit and the grating width further downstream narrow the beam divergence to less than the diffraction angle. The grating is machined into a thin SiN_xmembrane and has a period of d ¼100 nm. To minimize the dispersive van der Waals inter- action between the molecules and the grating wall we reduce the grating thickness from 160 nm (as in earlier diffraction experiments^5,20) to as little as 10 nm in our present set-up. This is important for the manipulation of complex molecules, which may exhibit high polarizabilities, permanent and even thermally induced electric dipole moments^26,27. Each individually diffracted molecule finally arrives at the 170-mm-thin quartz plate W₂, which seals the detector vacuum chamber against ambient air. The gradual emergence of the quantum interference pattern is then observed by means of wide- field fluorescence microscopy of W₂.

Imaging of single molecules in the condensed phase began about two decades ago²⁸, and various methods for subwavelength optical imaging have been developed since²⁹. Here, we make use of a scheme that is similar to single-molecule high-resolution imaging

1Vienna Center of Quantum Science and Technology, Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria,²The Center for Nanoscience and Nanotechnology, Tel Aviv University, 69978 Tel Aviv, Israel,³Department of Chemistry, University of Basel, St. Johannsring 19, 4056 Basel, Switzerland,⁴Karlsruhe Institute of Technology, Institute for Nanotechnology, PO Box 3640, 76021 Karlsruhe, Germany,⁵School of Chemistry, The Raymond and Beverly Sackler faculty of exact sciences, Tel Aviv University, 69978 Tel Aviv, Israel.*e-mail: markus.arndt@univie.ac.at

LETTERS

PUBLISHED ONLINE: 25 MARCH 2012 |DOI: 10.1038/NNANO.2012.34

NATURE NANOTECHNOLOGY| ADVANCE ONLINE PUBLICATION | www.nature.com/naturenanotechnology 1

X

[26]

... ftalocyanin:

[28]

Neutroner, He-atomer, C

₆₀

-molekyler och...

(STM-bilder)

FSS

(28)

Heisenbergs obestämdhetsrelation

8 Heisenberg (1927):

Omöjligt att bestämma en partikels läge och rörelsemängd samtidigt!

Δp

_x

⋅ Δx ≥ h 4π

oskärpa i rörelsemängd

oskärpa i läge

[29]

[30]

(29)

Heisenbergs obestämdhetsrelation

8 Heisenberg (1927):

Omöjligt att bestämma en partikels läge och rörelsemängd samtidigt!

Δp

_x

⋅ Δx ≥ h

4π Ex: Ljus genom enkelspalt

oskärpa i läge

Δx

Δp

_x

p

[29]

[30]

Använd fotonmodellen och betrakta en foton i spaltöppningen. Fotonens rörelsemängd är

Minska spaltbredden

→ mindre Δx → större Δp

_x

→ mer utsmetad intensitetsfördelning

p 

(30)

Heisenbergs obestämdhetsrelation

8 Heisenberg (1927):

Omöjligt att bestämma en partikels läge och rörelsemängd samtidigt!

Δp

_x

⋅ Δx ≥ h

4π Ljus genom enkelspalt

Använd fotonmodellen och betrakta en foton i spaltöppningen. Fotonens rörelsemängd är

Minska spaltbredden

→ mindre Δx → större Δp

_x

→ mer utsmetad intensitetsfördelning Ex:

oskärpa i läge

Δx

Δp

_x

p

Jfr vågmodellen:

[29]

[30]

p 

(31)

Läget runt 1927

9 Partikelegenskaper Vågegenskaper

Materia

Strålning

Ja! ^(mekanik)

p = mv

W

_k

= mv

²

2

p =

γ

^mv ^E_k = (

γ

−1)mc² R = p

Ja! (fotonmodellen)

!#5 V per fringe".

[9]

Ja! (elektromagnetism, optik)

W = hf

p = h

λ

^{E, B}

Ja! (kvantmekanik) i ∂ψ

∂t = − 

²

2m

∂

²

ψ

∂x

²

+U ψ

FSS

Dubbelspaltexperiment med fotoner (2005)

λ

= h p

(32)

Efter 1927?

10

2000 2010 2020 2030 2040 1900

1890 1910 1920 1930 1940

1880 1950 1960 1970 1980 1990

Tidig kvantmekanik Kvantfysik

Kvantkemi, materialfysik

Atomfysik, kärnfysik, partikelfysik

QED QCD

Relativistisk kvantmekanik, kvantfältteori

FSS

(33)

Efter 1927?

X

FSS

Efter 1927?

10 2000 2010 2020 2030 2040 1900

1890 1910 1920 1930 1940

1880 1950 1960 1970 1980 1990

Tidig kvant- mekanik Kvantfysik

Kvantkemi, materialfysik

Atomfysik, kärnfysik, partikelfysik

QED QCD

Relativistisk kvantmekanik, kvantfältteori

FSS

Vägintegralformulering (Feynman) Matrismekanik (Heisenberg, Dirac)

Vågmekanik (Schrödinger)

(34)

Efter 1927?

10

2000 2010 2020 2030 2040 1900

1890 1910 1920 1930 1940

1880 1950 1960 1970 1980 1990

Partikelfysikens standardmodell Kvantkemi, materialfysik

QED QCD

[31] [32]

FSS

(35)

Efter 1927?

X

2000 2010 2020 2030 2040 1900

1890 1910 1920 1930 1940

1880 1950 1960 1970 1980 1990

Partikelfysikens standardmodell Kvantkemi, materialfysik

QED QCD

[31] [32]

FSS

(36)

11 Läget idag

[33]

FSS

Ett sätt at se på det:

(37)

!#5 V per fringe".

[9]

Dubbelspaltexperiment med fotoner (2005)

Läget idag

12 Partikelegenskaper Vågegenskaper

Materia

Strålning

Ja! ^(mekanik)

p = mv

W

_k

= mv

²

2

p =

γ

^mv ^E_k = (

γ

−1)mc² R = p

Ja! (fotonmodellen) Ja! (elektromagnetism, optik)

W = hf

p = h

λ

^{E, B}

Ja! (kvantmekanik) i ∂ψ

∂t = − 

²

2m

∂

²

ψ

∂x

²

+U ψ

Kvantfältteorier

QED

QCD Partikelfysikens standardmodell

FSS

(38)

Källor

[0] http://researcher.ibm.com/researcher/view_project_subpage.php?id=4252 Fe-atomer på Cu(111).

[0a] http://www.farnamstreetblog.com/2015/01/richard-feynman-knowing-something/

[0b] Richard P. Feynman i The Character of Physical Law (Penguin Books, 1992) s. 129 [1] http://en.wikipedia.org/wiki/File:Richard_Feynman_Nobel.jpg

[1b] http://www.davidrumsey.com/luna/servlet/detail/RUMSEY~8~1~30692~1150611:Europe- Från The Times atlas, 1895

[1b2] http://www.massingnickel.se/bussar.html

[1c] T. L. Dimitrova & A. Weis, American Journal of Physics 76 (2008) 137 se också http://photonterrace.net/en/photon/duality/

solvay Se ockås http://www.youtube.com/watch?v=8GZdZUouzBY

[2] http://en.wikipedia.org/wiki/File:Broglie_Big.jpg

[3] http://www.diyphysics.com/2013/04/02/in-memoriam-dr-akira-tonomura-1942-2012/

[4] http://en.wikipedia.org/wiki/File:Erwin_Schrödinger.jpg [5] http://en.wikipedia.org/wiki/File:Phthalocyanine-3D-balls.png [6]

[7]

[8]

[9]

[10]

Vidareläsning

Kapitel 4 i Ett utsökt universum

av Brian Greene Quantum – A Guide for The Perplexed

av Jim Al-Khalili

i  ∂∂ t = − 2  m 2 ∂ ∂ 2 x 2 + U

Kvantfysik

Uppdaterad: 171211

[1] Läget runt 1920

[2] Materiens vågegenskaper (de Broglie)

[3] Men kan det vara så här? (Elektroner och gitter)

[4] Men kan det vara så här? (Elektroner och dubbelspalt) [5] Materiens vågegenskaper (Schrödinger – Born)

[6] Kvantmekanik (tidig) [7] Kvantmekanik (tidig)

[8] Heisenbergs obestämdhetsrelation [9] Läget runt 1927

[10] Efter 1927 [11] Läget idag

[12] Läget idag i  ∂ ψ

∂t = − 

2m

∂

ψ

∂x

+U ψ

Kvantfysik

X

Märklig och ointuitiv – men inte ologisk!

FSS

Några milstolpar i fysiken

Newton (Principia)

Young (Dubbelspaltexpt.) Maxwell (E och M)

Thomson (elektronen) Planck (energikvanta) Einstein (ljusenergi

kvantiserad) Bohr (ny atommodell)

1687

~1803

~1864 1897 1900 1905

1913 Spec. rel.teori

X

Några milstolpar i fysiken

Newton (Principia)

Young (Dubbelspaltexpt.) Maxwell (E och M)

Thomson (elektronen) Planck (energikvanta) Einstein (ljusenergi

kvantiserad) Bohr (ny atommodell)

1687

~1803

~1864 1897 1900 1905

1913 Spec. rel.teori

X

Läget runt 1920

X

FSS

Läget runt 1920

1

Partikelegenskaper Vågegenskaper

Materia

Strålning

Ja! (mekanik)

W

= mv

2

γ

γ

Ja! (fotonmodellen)

?

Ja! (elektromagnetism, optik)

λ

Dubbelspaltexperiment med fotoner (2005)

FSS

Läget runt 1920

X

FSS

Materiens vågegenskaper

2

de Broglie (1924):

Varje partikel kan tillskrivas en våglängd λ = h p

“materievåglängd”

Materiens vågegenskaper

Ger en slags förklaring till

väteatomens kvantiserade energinivåer:

2

de Broglie (1924):

Varje partikel kan tillskrivas en våglängd λ = h p

“materievåglängd”

2 π r = n λ

Materiens vågegenskaper

Ger en slags förklaring till

väteatomens kvantiserade energinivåer:

Ja! ^(mekanik)