qm1_20Figure 1.1: L
! = c ! n! = L (n = 1, 2, . . . )
. 3 d3" 2d3n = 2
L3
c3 "2d" (1.1)
2 T , 1 kBT ,
$(")d" = 8#
• ! T , , !
T • !! 0 $d" = ! (ultraviolet catastrophe).
1L. Rayleigh, Phil. Mag. 29, 539 (1900); J. Jeans, Phil. Mag. 10,91 (1905)
4
",h" nh", (n = 0, 1, 2, . . . ) (1.3)

2M. Planck, Verh. Deutsch. Phys. Ges. 2, 207, 237 (1900)
5
&$(")
3(ex " 1)" xex = 0 (1.10)
x = 2.82144, " = 2.821kBT h
! = c
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Planck hPlanck h = 6.62607015# 10"34J s (1.12)
6
Figure 1.3: D.J.Fixsen et al. Astrophys. J. 473, 573 (1996).
1.1.2 Bohr Rutherford Thomson (1904)4: (1904)5: ,
Geiger-Marsden (1909)6: ', ' Rutherford (1911)7: ', Rutherford
3Penzias, A. A. Wilson, R. W., A Measurement of Excess Antenna Temperature at 4080 Mc/s, As-
trophysical Journal, vol. 142, p.419-421, 1965 4J.J. Thomson, Philosophical Magazine Series 6, vol. 7 (39) p.237-265, 1904 5H. Nagaoka, LV. Kinetics of a system of particles illustrating the line and the band spectrum and
the phenomena of radioactivity, Phil. Mag. 7 (1904) 41 6H. Geiger and E. Marsden, Proc. Roy. Soc. A82, 495 (1909) 7E. Rutherford, Phil. Mag. 21,669 (1911)
7
• Balmer 18858
n2
% (1.17)
8J.J. Balmer, Notiz uber die Spectrallinien des Wassersto!s, Ann. der Phys. und Chem. 25, 80 (1985) 9T. Lyman, The spectrum of Hydrogen in the region of extremely short wave-length. Memoirs of the
American Academy of Arts and Sciences, New Series 13 (3): 125-146, 1906 10F. Paschen, Zur Kenntnis ultraroter Linienspaktra I Ann. der Phys. 332 13, 537-570, 1908 11F.S. Bracket, Visible and infrared radiation of hydrogen, Astrophysical Journ. 56, 154 (1922) 12A.H. Pfund, The emission of nitrgen and hydrogen in infrared, J. Opt. Soc. Am. 9(3), 193-196
(1924)
8
" = Em " En
h (1.19)
m v2
rB = 4#(0!2 me2
(1.26)
13N. Bohr, Phil. Mag. 26 1, 476, 857 (1913); Nature 92, 231 (1913)
9
, , 1.2 , ", ()E
, , Einstein, 1905 ", ( = h" 16,
, , 14J. Franck and G.L. Hertz, Verhandlungen der Deutschen Physikalischen Gesellschaft”, 16, 1914,
p.457-467. - Texte date de Berlin, le 5 mai 1914 15P. Lenard,“ Uber die lichtelektrische Wirkung”, Annalen der Physik 8, 149-198 (1902) 16A. Einstein, Ann. Phsik, 17, 132 (1905)
10
1.5. 3 # 10"7m 2.1eV 5# 10"7m 0.5eV Planck Planck Planck
! = h
1.3 Compton , ,
(Thomson)
18 .
17R.A.Millikan, Phys. Rev. 7,355 (1916) 18A. H. Compton, Phys. Rev. 21, 207 (1923)
11
( = h" = !* (1.29)
!k = !k# + p (1.31)
h
X 10"10m, 12
S1
S2
Figure 1.7: Young 2
1.5 de Broglie de Broglie() 192320, p, E
E = !*, p = !k (1.36)
*, k, 2 1.1. V ,
p2
2m = eV (1.37)
19, (, ) 20L. de Broglie, Nature 112 (1923) 540;A tentative theory of light quanta 1924; Recherches sur la
theorie des quanta, Annales de Physique 10, 2,1925
13
! = h
p =
2 2, 24 ( 1.8) 2 25
1.6 Review of Particle Data Group26
• c = 299792458 m s"1
• e = 1.602176634# 10"19 C
• me = 0.51099895000(15) MeV/c2 = 9.1093837015(28)# 10"31 kg
21C. Davisson and L. Germer, Phys. Rev. 30, 707 (1927) 22S.Kikuchi: Japan J. Phys. 5 (1928) 83. 23G.P.Thomson, Proc. Roy. Soc. A117, 600 (1928) 24C. Jonsson, Z. Phyzik 161 (1961) 454;
P.G. Merli, G.F. Missiroli, and G. Pozzi,“Di!razione e interferenza di elettroni. II-Interferenza,”Giornale di Fisica17 (1976), 83-101, on 89;American Journal of Physics 44, 306 (1976) https://www.hitachi.co.jp/rd/research/materials/quantum/doubleslit/index.html
25A.Zeilinger, Rev. Mod. Phys. 72 (1999) S288 26Prog. Theor. Exp. Phys. 8 (2020)
14
15
Figure 1.9: , A. Zeilinger, Rev. Mod. Phys. 72 (1999)S288 .
• mp = 938.27208816(29) MeV/c2 = 1.67262192369(51)# 10"27 kg
• Planck h = 6.62607015# 10"34 J s
• Planck ! = 1.054571817 · · ·# 10"34 J s
• Boltzman kB = 1.380649# 10"23 J K"1
• (0 = 8.8541878128(13)# 10"12 F·m"1
1eV = 1.602176634# 10"19J (1.39)
1J = 6.241509074# 1018eV (1.40)
mec 2 = 0.511MeV (1.43)
16

1900 Planck Planck 1905 Einstein 1909 Geiger and Marsden ' 1911 Rutherford 1913 Bohr 1914 Frank and Hertz 1916 Millikan h 1922 Compton 1923 de Broglie 1924 Pauli 1927 Heisenberg 1927 Davisson and Germer 1927 Born 1928 Dirac
17
p, E, de Broglie
! = h
k = 2#
2m
&2
E = p2
$ " !2 2m
E $ i! & &t
,(x, t) = A exp [i(k · x" *t)] (2.16)
, V (x, y, z) E =
p2
2m + V (x, y, z) (2.17)
27E. Schrodinger, Ann. der Phys. 79, 361, 489 (1926); 80, 437 (1926); 81,109 (1926)
19
&t , p $ "i!& (2.18)
Laplacian Schrodinger (2.13) 1, x 2 , ,1, ,2 Schrodinger , a1, a2 a1,1 + a2,2 Schrodinger t 1, t = 0, ,
,(x, 0), t 2 , 1 2.1. Schrodinger (2.13) 2
,1(x, t) = A1 exp[i(k1x" *1t)], p1 = !k1, E1 = !*1 = p21 2m
(2.21)
,2(x, t) = A2 exp[i(k2x" *2t)], p2 = !k2, E2 = !*2 = p22 2m
(2.22)
2.2.
2.2 , ”
|,|2” , Born Born 28 ,(x, t) 2
|,(x, t)|2d3x (2.25)
#
(normalizable) |,(x, t)|2 x , |,(x, t)|2
2.3. Schrodinger (2.16)
|,(x, t)|2 = |A|2 (2.27)
, .
# |,(x, t)|2d3x = |A|2! (2.28)
, 28M. Born, Z.Phyzik 37, 863 (1926); Nature 119,354 (1927)
21
2.4. 2.1 2, x |,|2 = |,1 + ,2|2
= |,1|2 + |,2|2 + ,1, $ 2 + ,$
1,2
i((k1"k2)x"i(%1"%2)t) + A$ 1A2e
i(("k1+k2)x"i("%1+%2)t) (2.29)
, 2.3 2 Born, ACAC 2
B1, x, P (x) )B 1 )1 2
) = )1 + )2 (2.30)
P (x) P (x) = |)|2 = |)1 + )2|2 (2.31)
2 P1 P1 = |)1|2 , 1 P2 P2 =
|)2|2 P '= P1 + P2 (2.32)
1 1 2 C P1 P2
22
S1
# |,(x, t)|2d3x (2.33)
d
dt
&
2m
$ , &2,
&x2 " ,$&
2,$
&x2
&$
,
$ = |,|2 = |A|2 (2.42)
= |A|2v (2.43)
# b
a
x = b x = a
24
&t =
2m
3&
$ ,$ &,
,(x, t) = A exp
j(x, t) = " i! 2m
26
# !
P (x) = 1% 2#-2
exp
$ "(x"m)2
2-2
% (3.4)
27
# !
# !
1 '
2#
3.1. f(x) = Ae"ax2 (a > 0) Fourier
f(k) =
1 '
2" &2k2
e" x2
• x '= 0, f(x) $ 0
. 0 x '= 0 ! x = 0
(3.20)
/(g(x)) = &
n
1
|g#(xn)| /(x" xn) (3.27)
g(x) x = xn (g(xn) = 0, g#(xn) '= 0) /(x2 " a2) =
1
30
# !
x/#(x) = "/(x) (3.31)

delta Fourier
31
,(x, t) , x * x + dx |,(x, t)|2dx x
+x, = # !
# !
"! ,$(x, t)f(x),(x, t)dx (3.36)
x ("x)2 ("x)2 = +(x" +x,)2, = +(x2 " 2x+x,+ +x,2),
= +x2, " +x,2 (3.37)
p p * p+ dp Fourier
,(x, t) =
#
+p, =
#
&x (3.48)
33
. Axe"&x x > 0
0 x < 0 (3.51)
,
1. A 2. $(x) = |)(x)|2 3. x +x, ("x)2 4. )(p) 5. +p, ("p)2
A(x, p), A = A(x,"i! &
&x ) (3.52)
3.2. Hamlitonian
J = x# ("i!&) (3.59)
A(x,"i! $
$x) x, t,(x, t) , (x, t) , B , A, AB ,
(AB), = A(B,) (3.60)
A, B AB AB BA 3.4. x p xp px
xp, = x("i!)&, &x
[A, B] = AB " BA (3.63)
AB BAx p [x, x] = 0, [p, p] = 0 (3.64)
[x, p] = i! (3.65)
3.2. [H, x], [H, p] 3.3. [Lx, Ly], [Ly, Lz], [Lz, Lx]
35
d
36
&x , (3.72)
x f(x) f(x) +f(x), x +x, f(+x,)
F (x) = "dV (x) dx +x,
F (x) = F (+x,) + F #(+x,)(x" +x,) + d2F (+x,) d+x,2 (x" +x,)2 + · · · (3.73)
, +F (x), = F (+x,) + 1
2
x, p 000000
d2F ('x() d'x(2 ("x)2
$ iS(x, t)
i!&, &t
"A
&t = " 1