Quartz sync: Mar 20, 2024, 3:59 PM
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@@ -3,11 +3,33 @@
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> [!question] Frage
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> Wie funktioniert das Franck-Hertz-Experiment, und welche Bedeutung hat es für die Quantenmechanik?
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>
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> Weitere Fragen:
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> - Worin besteht der Unterschied zum Photoeffekt?
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> - Welches Elektron im $\ce{Hg}$-Gas wird angeregt und wie ist der Übergang?
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> - In welche Raumrichtungen geschieht die Emission?
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> - Warum treffen die Elektronen nicht auf die Gitteranode?
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> - Wie funktioniert die Glühkathode (inkl. Verteilung der Elektronen)?
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# Antwort
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![[Franck-Hertz-Versuch]]
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## Weitere Fragen
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### Worin besteht der Unterschied zum Photoeffekt?
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Beim [[Der photoelektrische Effekt|Photoeffekt]] wurde die Quantisierung der [[Photon]]enenergie in Proportionalität zu deren Frequenz festgestellt, während der *Frank-Hertz-Versuch* die Quantisierung der Energieaufnahme von Atomen festgestellt hat.
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### Welches Elektron im Hg-Gas wird angeregt und wie ist der Übergang?
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Die Valenzelektronen beim Quecksilber sind in der 6s-Schale ([Xe] 4f¹⁴ 5d¹⁰ 6s²) und gehen in die p-Schale über mit einer Energiedifferenz von $\pu{4.9 eV}$.
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### In welche Raumrichtungen geschieht die Emission?
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Der Raum besitzt keine Vorzugsrichtung. Die Abstrahlung ist isotrop.
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### Warum treffen die Elektronen nicht auf die Gitteranode?
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Das elektrische Feld, welches die Elektronen in Richtung der Gitteranode beschleunigt ist weitestgehend homogen bis kurz vor der Anode, weshalb sie gleich wahrscheinlich in Richtung einer Gitterlücke beschleunigt werden. Es sehr nah am Gitter wird das Feld inhomogen, wobei die Elektronen jedoch aufgrund ihrer Trägheit nicht beeinflusst werden ($\pu{4,9 eV} \equiv \pu{1.3e6 m/s}$).
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Trotz dessen kommt es zu einem Strom an Elektronen, welche in die Gitteranode eintreten. Dieser weißt selbst keinen Franck-Hertz-Verlauf auf, da es zu keinem Energiecutoff aufgrund einer Bremsspannung kommt.
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### Wie funktioniert die Glühkathode (inkl. Verteilung der Elektronen)?
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Die Glühkathode wird erhitzt wobei die Elektronen durch die [[Stochastische Verteilungen#*Fermi-Dirac*-Verteilung]] beschrieben werden können. Einige Elektronen besetzen hierdurch ab einer gewissen Temperatur auch ionisierte Zustände und werden ausgelöst.
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![[fermi_dirac_glühkathode.png]]
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# Vorkommen
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1. [[Notizen_2023-12-21_Altprotokoll_Wolf]]
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@@ -6,11 +6,10 @@
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>
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# Antwort
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![[Laser]]
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# Vorkommen
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1. *Insert link here*
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2. *And more*
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1. *Nie*
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*Folder tags:*
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@@ -9,6 +9,12 @@
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![[Der photoelektrische Effekt]]
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> [!question] Alternative Frage
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> Erläutere den Photoeffekt und beschreibe die beiden relevanten Kennlinien, die mit diesem Phänomen verbunden sind. Was sagt der y-Achsenabschnitt bei der Energie-Frequenz-Kennlinie aus?
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>
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Beim Photoeffekt werden wird monochromatisches Licht auf einen Festkörper geschiehenen. Man beobachtet eine Auslösung der Elektronen und eine obere Grenze für die kinetische Energie der Elektronen, welche nicht von der Intensität sondern nur von der Frequenz abhängt. Zur Bestimmung der Elektronenenergie wird eine Gegenspannung angelegt, wobei die Spannung notiert wird, bei welcher keine Elektronen mehr eintreffen. Wenn dieser Versuch für verschiedene Frequenzen durchführt erhält man eine Gerade mit Steigung des Planckschen Wirkungsquantums. Der y-Achsenabschnitt gibt die Auslösearbeit an.
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# Vorkommen
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1. [[Notizen_2023-06-06_Altprotokoll_Wolf]]
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2. [[Notizen_2023-12-21_Altprotokoll_Wolf]]
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@@ -12,7 +12,7 @@ $$
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\left(\frac{\mathrm{d}E}{\mathrm{d}x}\right)_{ion} = \left(\frac{\mathrm{d}E}{\mathrm{d}x}\right)_{brems}
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$$
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mit $E_c \approx \frac{610\,\mathrm{MeV}}{Z+1.24}$.
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Des Weiteren weißt das Bremsstrahlungs-Spektrum diskrete [[Röntgenstrahlung|Röntgen-Peaks]] auf.
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Des Weiteren weißt das Bremsstrahlungs-Spektrum bei einer [[Röntgenstrahlung|Röntgenröhre]] diskrete [[Röntgenstrahlung|Röntgen-Peaks]] auf.
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*Folder tags:*
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@@ -1,6 +1,6 @@
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#class/Ex-IV #experiment
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*Albert Einstein* erklärte 1905 den photoelektrischen Effekt, der zeigt, dass Licht aus Quanten, den sogenannten [[Photon]]en, besteht. Diese Entdeckung, die durch Experimente mit der Gegenfeldmethode quantitativ belegt wurde, unterstrich den Teilchencharakter des Lichts und führte zu Einsteins Nobelpreis im Jahr 1921.
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*Albert Einstein* erklärte 1905 den photoelektrischen Effekt ([[Photoeffekt]]), der zeigt, dass Licht aus Quanten, den sogenannten [[Photon]]en, besteht. Diese Entdeckung, die durch Experimente mit der Gegenfeldmethode quantitativ belegt wurde, unterstrich den Teilchencharakter des Lichts und führte zu Einsteins Nobelpreis im Jahr 1921.
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@@ -12,6 +12,11 @@ Die experimentelle Untersuchung des photoelektrischen Effekts basiert auf der Be
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*Albert Einstein* lieferte 1905 eine Erklärung für diese Beobachtungen, indem er die Quantisierung von Licht vorschlug. Er erhielt für diese Erklärung, die Licht als aus Photonen bestehend annimmt, 1921 den Nobelpreis. Jedes Photon trägt die Energie $E = h\nu$, wobei $\nu$ die Frequenz des Lichtes ist. Der photoelektrische Effekt lieferte somit einen direkten Beweis für die Quantennatur des Lichts und spielte eine zentrale Rolle in der Entwicklung der Quantenmechanik.
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![[photoeffekt-strom-spannung.png]]
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Im Experiment zum photoelektrischen Effekt zeigt das Strom-Spannungs-Diagramm deutlich, wie die Lichtintensität die Anzahl der emittierten Elektronen und somit den Sättigungsstrom beeinflusst. Mit steigender Lichtintensität erhöht sich die Anzahl der Photonen, die auf das Material treffen, was zu einer größeren Anzahl von emittierten Elektronen führt. Dies äußert sich in einem höheren Sättigungsstrom für höhere Lichtintensitäten, wie in den Kurven dargestellt. Die Sättigung zeigt, dass alle verfügbaren Photoelektronen von der Anode eingefangen werden.
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Der Stoppspannung, der Spannung, bei der der Strom auf null fällt, weil selbst die energiereichsten Photoelektronen nicht mehr die Anode erreichen können, bleibt jedoch unabhängig von der Lichtintensität konstant. Dies liegt daran, dass die Stoppspannung nur von der Energie der Photonen (ihrer Frequenz) und der Austrittsarbeit des Materials abhängt, nicht aber von der Anzahl der Photonen.
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*Folder tags:*
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@@ -10,7 +10,7 @@
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- [ ] Altprotokoll vom 2023-02-09: [[Notizen_2023-02-09_Altprotokoll_Wolf|Link zum Protokoll]]
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- [ ] Altprotokoll vom 2022-11-07: [[Notizen_2022-11-07_Altprotokoll_Wolf|Link zum Protokoll]]
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- [ ] Altprotokoll vom 2022-10-19: [[Notizen_2022-10-19_Altprotokoll_Wolf|Link zum Protokoll]]
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- [ ] Altprotokoll vom 2022-09-08: [[Notizen_2022-09-08_Altprotokoll_Wolf|Link zum Protokoll]]
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- [x] Altprotokoll vom 2022-09-08: [[Notizen_2022-09-08_Altprotokoll_Wolf|Link zum Protokoll]]
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- [ ] Altprotokoll vom 2022-09-00: [[Notizen_2022-09-00_Altprotokoll_Wolf|Link zum Protokoll]]
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- [ ] Altprotokoll vom 2019-12-19: [[Notizen_2019-12-19_Altprotokoll_Wolf|Link zum Protokoll]]
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- [ ] Altprotokoll vom 2019-04-18: [[Notizen_2019-04-18_Altprotokoll_Wolf|Link zum Protokoll]]
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@@ -17,6 +17,7 @@ $$E = a_v A - a_o A^\frac{2}{3} - a_c Z (Z-1) A^{-\frac{1}{3}} + a_s \frac{(N-Z)
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0 &\text{für } \mathbf{g,u} \text{ oder } \mathbf{u,g}\\
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-a_p A^{-\frac{1}{2}} &\text{für } \mathbf{u,u}
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\end{cases}$$
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![[Bethe-Weizsäcker.svg]]
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Diese Korrekturen favorisieren symmetrische Kombinationen von [[Proton]]en und [[Neutron]]en sowie eine gerade Anzahl beider Nukleonenarten. Der Term $a_s$ bevorzugt symmetrische Kerne, während $a_p$ gerade Anzahlen von Protonen und Neutronen begünstigt.
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@@ -21,6 +21,7 @@ Außerdem werden kurz angerissen:
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$$\ce{^A_ZA} \longrightarrow \ce{^{A}_{Z-1}B} + {\mathbf{e}^+}+ \nu_e$$
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## Elektroneneinfang
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![[Beta Plus Decay.svg]]
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$$\ce{^Z_AA} + {\mathbf{e}^-}\longrightarrow \ce{^A_{Z-1}B} + \nu_e$$
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@@ -12,6 +12,7 @@
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Das Spektrum besteht aus einem kontinuierlichen Spektrum der [[Bremsstrahlung]] und charakteristische Peaks aufgrund von Übergängen in die *K*-Schale ($n=1$) aus angehobeben Zuständen.
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![[Röntgenspektrum-Molybdän_Energien.svg]]
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Das [[Bremsstrahlung]]sspektrum weißt einen Cutoff auf, welcher die maximale Elektronenenergie repräsentiert (mehr Energie kann nicht in ein Photon umgewandelt werden)
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*Folder tags:*
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@@ -1,7 +1,7 @@
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#class/Ex-IV #kerne
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## Streuquerschnitt und Übergangswahrscheinlichkeit
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Der Streuquerschnitt wird definiert als $\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}$, wobei die einlaufende Welle als $\psi_e = e^{i \vec{k} \vec{r}}$ und die auslaufende Welle als $\psi_a = e^{i \vec{k}' \vec{r}}$ angenommen werden. Die Übergangswahrscheinlichkeit, formuliert durch das Streupotential $V(r)$, ist
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Der Streuquerschnitt wird definiert als $\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega}$, wobei die einlaufende Welle als $\psi_e = e^{i \vec{k} \vec{r}}$ und die auslaufende Welle als $\psi_a = e^{i \vec{k}' \vec{r}}$ angenommen werden. Die Übergangswahrscheinlichkeit, formuliert durch das Streupotential $V(r)$, ist (vgl. [[Fermis Goldene Regel]])
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$$M_{fi} = \left\langle \psi_a \middle\lvert V(r) \middle\rvert \psi_e \right\rangle$$
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@@ -21,9 +21,9 @@ Das Streupotential einer Punktladung wird durch das Coulomb-Potential repräsent
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Die Streuung an einer beliebigen Ladungsverteilung führt zu einem Streupotential, das sich aus einer Faltung der Ladungsverteilung mit dem Coulomb-Potential ergibt. Dies resultiert in einem Streuquerschnitt, der ebenfalls proportional zu $\frac{1}{q^4} \left\lvert \rho(q) \right\rvert^2$ ist, wobei $\rho(q)$ die Fouriertransformierte der Ladungsdichte darstellt.
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Für eine konstante Ladungsdichte innerhalb einer Kugel und deren Verschwinden außerhalb ergibt sich eine Besselfunktion als Lösung. Dies unterstützt die Annahme einer gleichmäßigen Protonenverteilung im Kern und definiert einen festen Kernradius:
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Für eine konstante Ladungsdichte innerhalb einer Kugel und deren Verschwinden außerhalb ergibt sich eine Besselfunktion als Lösung. Dies unterstützt die Annahme einer gleichmäßigen Protonenverteilung im [[Kernmodelle|Kern]] und definiert einen festen Kernradius:
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$$R = \pu{{1.128} {fm}} A^{\frac`{1}{3}}$$ mit $A = N + Z$. Dieses Modell unterstützt die Vorstellung eines kompakten Kerns mit einer konstanten Ladungsdichte im Zentrum von ca. $\pu{{.17} {per cubic femto m}} \frac{Z e}{A}$.
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$$R = \pu{{1.128} {fm}} A^{\frac`{1}{3}}$$ mit $A = N + Z$. Dieses Modell unterstützt die Vorstellung eines kompakten Kerns mit einer konstanten Ladungsdichte im Zentrum von ca. $\pu{{.17} {per cubic femto m}} \frac{Z e}{A}$.
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*Folder tags:*
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@@ -12,6 +12,16 @@ $$
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Die *Fermi-Energie* ist die dem Impuls zugeordnete Energie. Ein solches Gas wird auch als **entartetes Gas** bezeichnet.
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Im Extremfall von $T=0$ sind alle Zustände bis $p_F$ besetzt und kein einziger darüber. Mit steigender Temperatur verschwimmt die Grenze. Für die Verteilung siehe [[Stochastische Verteilungen#*Fermi-Dirac*-Verteilung]]
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# Ladungstransport
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Der Ladungstransport im **Elektronengas** wurde zuerst von *Paul Drude* im **Drude-Modell** erklärt. Die Elektronen können sich frei bewegen. Es entsteht ein Widerstand durch Kollision mit den Atomrümpfen, wobei die elektrische Leitfähigkeit abgeleitet werden kann als
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$$
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\sigma = \frac{n e^{2}}{m_{e}} \tau
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$$
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mit der *mittleren freien Flugzeit* $\tau$.
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*Arnold Sommerfeld* erweitert das Modell um die quantenmechanische Betrachtung mit den Folgen aus dem Abschnitt [[#Fermi-Gas]] (Stichwort [[Stochastische Verteilungen#*Fermi-Dirac*-Verteilung]], [[Pauli-Prinzip]], usw.).
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*Folder tags:*
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@@ -18,6 +18,11 @@ Eine einfachere Überprüfung dieser Bedingung ist über das [[Reziprokes Gitter
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Weitere mathematischen Methoden zur Bestimmung der Reflexionsrichtungen sind die [[Ewald-Kugel]] und die [[Brillouin-Zone]].
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# Debye-Scherrer-Verfahren
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Zur Untersuchung von kristallinen Pulvern wird gerne das *Debye-Scherrer-Verfahren* verwendet, bei welchem zusätzlich zu den Einkristallen auch zufällige Anordnungen ausgewertet werden können.
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Wenn die gestreute Strahlung aufgefangen wird, können glatte Ringe beoobachtet werden anstatt einfacher Punkte. Durch ausmessen der Ringe kann die [[Gittergeometrie]] bestimmt werden.
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![[Debye-Scherrer.png]]
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*Folder tags:*
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@@ -0,0 +1,22 @@
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#class/Ex-IV #QM
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Das **Korrespondenzprinzip** in der Quantenmechanik stellt eine Brücke zwischen der klassischen Physik und der Quantenphysik dar. Es besagt, dass die Gesetze der Quantenmechanik in den Grenzen großer Quantenzahlen oder bei Systemen mit großen Massen in die klassischen physikalischen Gesetze übergehen. Dieses Prinzip, ursprünglich von Niels Bohr formuliert, erlaubt es, klassische Konzepte und Intuitionen auf die Quantenwelt zu übertragen, allerdings mit gewissen Anpassungen.
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In der Quantenmechanik werden physikalische Größen durch Operatoren dargestellt, die auf Wellenfunktionen wirken, um messbare Eigenschaften (Observable) wie Energie, Impuls und Position zu beschreiben. Diese Operatoren sind mathematische Werkzeuge, die es ermöglichen, die Wahrscheinlichkeitsverteilungen und Erwartungswerte physikalischer Größen zu berechnen.
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Hier sind einige der wichtigsten Operatoren in der Quantenmechanik und ihre korrespondierenden Darstellungen in der klassischen Physik:
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1. **Position** $x$: Der Operator für die Position eines Teilchens ist einfach der Ort $x$. In der Schrödinger-Gleichung wird die Position direkt als Variable verwendet. Korrespondierend: $$x \rightarrow x$$.
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2. **Impuls** $p$: In der Quantenmechanik wird der Impuls eines Teilchens durch den Operator $\hat{p} = -i\hbar \frac{d}{dx}$ dargestellt, wobei $\hbar$ das reduzierte Plancksche Wirkungsquantum und $i$ die imaginäre Einheit ist. Dieser Operator wirkt auf die Wellenfunktion, um den Impuls des Teilchens zu ermitteln. Korrespondierend: $$$p \rightarrow -i\hbar \frac{d}{dx}$$.
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3. **Energie** (Hamilton-Operator) $H$: Der Hamilton-Operator beschreibt die Gesamtenergie eines Systems und in der Quantenmechanik setzt er sich oft aus dem kinetischen Energieoperator und dem potentiellen Energieoperator zusammen. Für ein Teilchen in einem Potential $V(x)$ lautet der Hamilton-Operator $\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)$, wobei $m$ die Masse des Teilchens ist. Korrespondierend: $$E \rightarrow -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)$$.
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4. **Drehimpuls** $L$: Der Drehimpulsoperator in der Quantenmechanik ist komplexer als in der klassischen Mechanik und hängt von der Winkelposition und dem Winkelimpuls ab. In kartesischen Koordinaten wird er durch $\hat{L} = -i\hbar(x \frac{d}{dy} - y \frac{d}{dx})$ für die z-Komponente des Drehimpulses repräsentiert. Korrespondierend: $$L \rightarrow -i\hbar(x \frac{d}{dy} - y \frac{d}{dx})$$.
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Diese Operatoren und ihre Anwendungen sind zentral in der Quantenmechanik und ermöglichen die Berechnung von Wahrscheinlichkeiten und Erwartungswerten für physikalische Größen. Das Korrespondenzprinzip hilft dabei, die Verbindung zwischen den quantenmechanischen Operatoren und den klassischen physikalischen Größen herzustellen, indem es zeigt, wie die quantenmechanischen Beschreibungen in den klassischen Grenzfall übergehen.
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*Folder tags:*
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#folder-Physikalische-Grundlagen #folder-Quantenmechanik
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@@ -11,7 +11,7 @@ $$- \frac{\hbar^2}{2m} \vec{\nabla}^2 \psi(r,t) = i \hbar \frac{\partial}{\parti
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# Korrespondenzprinzip und vollständige Schrödingergleichung
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Das Korrespondenzprinzip besagt, dass die Quantenmechanik in den Grenzfall klassischer Physik übergehen muss. Bei nicht verschwindendem Potential $V \neq 0$ wird dieses als Operator $\hat{V}(x) = V(x)$ betrachtet, was zur vollständigen Schrödingergleichung führt:
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Das [[Korrespondenzprinzip]] besagt, dass die Quantenmechanik in den Grenzfall klassischer Physik übergehen muss. Bei nicht verschwindendem Potential $V \neq 0$ wird dieses als Operator $\hat{V}(x) = V(x)$ betrachtet, was zur vollständigen Schrödingergleichung führt:
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$$\left(-\frac{\hbar^2}{2m} \vec{\nabla}^2 + V(x)\right) \psi(x,t) = i \hbar \frac{\partial}{\partial t} \psi(x,t).$$
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# Messung der Wellenfunktion
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@@ -0,0 +1,507 @@
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Reference in New Issue
Block a user