Wave function of a photoelectron and its collapse in the photoemission process Hiroaki Tanaka

2025-05-06 0 0 1006.19KB 8 页 10玖币

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Wave function of a photoelectron and its collapse in the photoemission

process

Hiroaki Tanaka∗

Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan

Based on the ﬁrst-order perturbation theory, we show that the wave function of a photoelectron

is a wave packet with the same width as the incident light pulse. Photoelectron detection mea-

surements revealed that the widths of signal pulses were much shorter than the light pulse and

independent of the origin (photoemission or other noises), which is an experimental observation

of the wave function collapse. Signal pulses of photoelectrons were distributed along the time axis

within the same width as the light pulse, consistent with the interpretation of a wave function as

a probability distribution.

Photoelectron

Light Observer

Photoemission Observation

Keywords: Photoelectric eﬀect, Angle-resolved photoemission spectroscopy

∗Corresponding author: hiroaki-tanaka@issp.u-tokyo.ac.jp

arXiv:2210.13913v1 [cond-mat.mtrl-sci] 25 Oct 2022

The photoelectric eﬀect, the emission of electrons due to the irradiation of ultraviolet light or

x-rays, was found in 1887 by Hertz [1,2]. The kinetic energy of the liberated electrons and the

light frequency dependence of the phenomenon cannot be explained by classical physics, and the

photoelectric eﬀect, as well as the black body radiation and the Compton scattering, invoked the

development of quantum physics [3]. The photoemission process has been utilized to investigate the

electronic structure of solids in the name of photoemission spectroscopy (PES) and angle-resolved

PES (ARPES). Since momenta and energies of photoelectrons are related to Bloch wavevectors and

binding energies in crystals, ARPES can experimentally determine the band dispersion of solids [4].

Owing to the surface sensitivity of ARPES [5], this technique has been utilized to investigate the

electronic structures of two-dimensional materials and crystal surfaces such as graphene [6,7] and

topological surface states [8].

The photoemission process has been described by the ﬁrst-order perturbation theory and the one-

electron approximation [2,4]. The perturbation theory treats the incident light as a perturbative

vector potential, and the energy conservation law is derived from Fermi’s golden rule. The matrix

element term in the equation of the excitation probability gives momentum conservation law. Here,

the ﬁnal state, the wave function of a photoelectron, is a plane wave in the vacuum and rapidly

decays into the bulk in the one-step model [4,9]; the photoelectron wave function inﬁnitely spreads

in the vacuum. On the other hand, the three-step model, more widely used than the one-step model,

describes a photoelectron as a classical particle generated in the bulk and escaping from the surface

[4]. Furthermore, time-of-ﬂight spectrometers measure the ﬂight time of a photoelectron to determine

its kinetic energy [10], where the photoelectron is treated as a particle. A photoelectron has been

depicted as both a wave and a particle depending on models and the concrete description has been

missing.

In this report, we discuss the time evolution of a photoelectron wave function based on the time-

dependent ﬁrst-order perturbation theory. We derive that the wave function is the same as a plane

wave only in the area determined by the incident light position, the momentum of the photoelectron,

and the time, which is consistent with classical intuition. The time width of the wave packet is

the same as the pulse width of the incident light. We experimentally generated 80 µs-long wave

packets using a single crystal of FeSe and a 4.66 eV CW laser and detected them by a channel

electron multiplier (CEM). The photoelectron signal was shorter than 10 ns, which means that the

wave function indeed collapses in the observation process by the CEM. We also demonstrate that

wave functions are interpreted as probability distributions from the spread of photoelectron signal

positions.

We consider a system like Figure 1to calculate a photoelectron wave function; A crystal is placed

in the x < 0 area, the incident light with frequency ωis irradiated in the N1a < y < N2aregion

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WavefunctionofaphotoelectronanditscollapseinthephotoemissionprocessHiroakiTanakaInstituteforSolidStatePhysics,TheUniversityofTokyo,Kashiwa,Chiba277-8581,JapanBasedontherst-orderperturbationtheory,weshowthatthewavefunctionofaphotoelectronisawavepacketwiththesamewidthastheincidentlightpulse.Photoele...

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Wave function of a photoelectron and its collapse in the photoemission process Hiroaki Tanaka

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