Chapitre 1 : AMPLITUDE DE PROBABILITE

!"#!"$%&'#&(% % )*+,-%."/0123415%6789:7;-%
%
!"#$%&'()*)+),-./01234)34).567,70/014)
%
*8 9:;<&%:;)=>:;=()#??:<%@()A)B;()$#'&%<BC()+)
%
)<%=-;+>?@A?:B%+:C@8-A-%=-%8D,A<A%=D7B-%@<>A?+78-%=-%C<;;-%C%=<B;%8D-;@<+-%E%8D?B;A<BA%A%;-%F<?A%<7%C:*-B%
=D7B-%F:B+A?:B%+:C@8-G-%HI𝑟JAK%I%L%@;?%MKN%
"-AA-%F:B+A?:B%=D:B=-%-;A%B:CC,-%#D$C%&B=()=()$':E#E%C%&@N%
%
)<%@>:O<O?8?A,%=-%A>:7P->%8<%@<>A?+78-%=<B;%7B%P:87C-%=Q%=-%8D-;@<+-%-;A%R%%
𝑑𝑝 =!Ψ(𝑟,t)!!.𝑑𝑉%
%
F8 3(;?%&@)C%;@%GB()=()$':E#E%C%&@8)
%
!:7>%7B%@>:O8SC-%7B?=?C-B;?:BB-8%I%B-%=,@-B=<BA%T7-%=-%G%K%8D<C@8?A7=-%=-%@>:O<O?8?A,%-;A%R%
HIGJAK%
%
)<%@>:O<O?8?A,%=-%A>:7P->%8<%@<>A?+78-%=<B;%7B-%>,U?:B%=G%=-%8D-;@<+-%-;A%R%%
𝑑𝑝 =!Ψ(𝑥,t)!!.𝑑𝑥%
%
)<%=-B;?A,%8?B,?T7-%=-%@>:O<O?8?A,%-;A%=:B+%R%%
𝑑𝑝
𝑑𝑥 =!Ψ(𝑥,t)!
!
%
%
)<%@>:O<O?8?A,%=-%A>:7P->%8<%@<>A?+78-%-BA>-%G%V%WX%-A%G%V%YX%-;A%=-%&J%<?B;?%R%%
Ψ(𝑥,t)!!.𝑑𝑥 =1
!!
!!
%
"-AA-%+:B=?A?:B%-;A%<@@-8,-%<:;=%&%:;)=();:'D#C%?#&%:;N%
%
%
H8 .'%;<%$()=()?B$('$:?%&%:;8)
%
3:7A-%+:CO?B<?;:B%8?B,<?>-%=-%F:B+A?:B;%=D:B=-%-;A%,U<8-C-BA%7B-%F:B+A?:B%=D:B=-%@:;;?O8-N%
%
1G@,>?-B+-%=-;%A>:7;%=DZ:7BU%<P-+%=-;%@9:A:B;J%@7?;%=-;%,8-+A>:B;%%I3:B:C7><%&[\[K%N%
9AA@R##>=UN-GAN9?A<+9?N+:N]@#>=#C:P?--#=:7O8-;8?A-N^CP%
9AA@R##^^^N*:7A7O-N+:C#^<A+9_PV4`Wa!b57A9+%

!"#!"$%&'#&(% % )*+,-%."/0123415%6789:7;-%
%
%
1G@->?-B+-%<P-+%=-;%C:8-+78-;%=-%@9A<8:+*<B?B-N%
9AA@R##^^^N*:7A7O-N+:C#^<A+9_F-<A7>-V@8<*->c=-A<?8@<U-dPVP"?e6f25gh2%
%
eB%+:B;?=S>-%7B-%-G@,>?-B+-%=D?BA->F,>-B+-;%=-%F-BA-;%=DZ:7BU%I<P-+%;:7>+-%87C?B-7;-%:7%=-%@<>A?+78-;%
C<A,>?-88-;K%-B%;D<@@7*<BA%;7>%8D?BA->@>,A<A?:B%@>:O<O?8?;A-J%+D-;A%E%=?>-%-G@8:?A<BA%8<%F:B+A?:B%=D:B=-N%
%
)-;%A>:7;%;:BA%A:7]:7>;%+:B;?=,>,;%?=-BA?T7-;%-A%L%,+8<?>,;%M%-G<+A-C-BA%=-%8<%CiC-%F<j:BN%
%
eB%B:A-%R%
k%H&%8<%F:B+A?:B%=D:B=-%<;;:+?,-%E%7B%T7<BA:B%<P-+%8-%A>:7%;7@,>?-7>%:7P->A%;-78%l%
k%Hm%8<%F:B+A?:B%=D:B=-%<;;:+?,-%E%7B%T7<BA:B%<P-+%8-%A>:7%?BF,>?-7>%:7P->A%;-78%l%
k%H%8<%F:B+A?:B%=D:B=-%<;;:+?,-%E%7B%T7<BA:B%<P-+%8-;%=-7G%A>:7;%:7P->A;N%
%
.?%;-78%8-%A>:7%;7@,>?-7>%-;A%:7P->AJ%<8:>;%8<%@>:O<O?8?A,%T7D7B%T7<BA:B%@<>P?-BB-%;7>%7B%,8,C-BA%=-%
;7>F<+-%=.%?BF?B?A,;?C<8%@8<+,%-B%6%;7>%8D,+><B%E%8D?B;A<BA%A%-;A%R%
=!&I6J%AK%V%nH&I6J%AKnm%=.%
%
.?%;-78%8-%A>:7%?BF,>?-7>%-;A%:7P->AJ%<8:>;%8<%@>:O<O?8?A,%T7D7B%T7<BA:B%@<>P?-BB-%;7>%7B%,8,C-BA%=-%
;7>F<+-%=.%?BF?B?A,;?C<8%@8<+,%-B%6%;7>%8D,+><B%E%8D?B;A<BA%A%-;A%R%
=!mI6J%AK%V%nHmI6J%AKnm%=.%
%
o%@>,;-BAJ%:B%:7P>-%8-;%=-7G%A>:7;%-A%8<%@>:O<O?8?A,%=-P?-BA%R%
=!I6J%AK%V%nHI6J%AKnm%=.%%
%
!<>%<B<8:U?-%<P-+%8<%;7@->@:;?A?:B%=-;%<C@8?A7=-;%=D:B=-;%+:9,>-BA-;%-B%:@A?T7-%-A%8-;%=-7G%A>:7;%]:7<BA%
=-;%>p8-;%;*C,A>?T7-;J%:B%@-7A%=:B+%@>:@:;->%8<%F:B+A?:B%=D:B=-%A:A<8-%R%
𝛹𝑀,𝑡=𝛹!𝑀,𝑡+𝛹!𝑀,𝑡%
%
q8:>;J%8<%@>:O<O?8?A,%=-%=,A-+A?:B%;7>%8D,+><B%=-P?-BA%
𝑑𝑝 =!!𝛹!𝑀,𝑡+𝛹!𝑀,𝑡!.𝑑𝑆%
𝑑𝑝 =!!𝛹!𝑀,𝑡!.𝑑𝑆+!𝛹!𝑀,𝑡!.𝑑𝑆 +!𝛹!𝑀,𝑡!𝛹∗
!𝑀,𝑡.𝑑𝑆 +𝛹∗ !𝑀,𝑡.!𝛹!𝑀,𝑡.𝑑𝑆%
%
%
1B%@:;<BA%R%
H&I6J%AK%V%nH&I6J%AKn%-]%r&I6K%
HmI6J%AK%V%nHmI6J%AKn%-]%rmI6K%
1A%%
srI6K%V%rmI6K%t%r&I6K%%
%
8-%=,@9<;<U-%-B%6%-BA>-%8-;%:B=-;%=-%@>:O<O?8?A,J%:B%:OA?-BA%R%
%
𝑑𝑝 =!!𝛹!𝑀,𝑡!.𝑑𝑆+!𝛹!𝑀,𝑡!.𝑑𝑆 +2!𝛹!𝑀,𝑡!
!!𝛹!𝑀,𝑡!.cos 𝛥𝜑 .𝑑𝑆%
;:?A%R%
%
𝑑𝑝 =!𝑑𝑝!+𝑑𝑝!+2𝑑𝑝!𝑑𝑝!.cos 𝛥𝜑 %
%
gB-%A-88-%8:?%+:>>-;@:B=%O?-B%<7G%:O;->P<A?:B;%R%8-;%=,A-+A?:B;%;-%>,@<>A?;;-BA%;7?P<BA%7B-%8:?%=-%
@>:O<O?8?A,%=:BA%8-%@>:F?8%-;A%+-87?%=-%8<%F?U7>-%=D?BA->F,>-B+-;%E%=-7G%:B=-;%=-%Z:7BUN%
%

!"#!"$%&'#&(% % )*+,-%."/0123415%6789:7;-%
%
%
!"#$%&'()F)+)4I2,106J)34)K!L5M30JN45).625)2J4).,510!2/4)/0754)
%
*8 4GB#&%:;)=()K<"':=%;O(')+)
%
)D,T7<A?:B%=-%.+9>u=?BU->%-;A%8D,T7<A?:B%>,U?;;<BA%8-;%P<>?<A?:B;%=-%8<%F:B+A?:B%=D:B=-%Ψ(𝑟,t)N%
"D-;A%7B%@:;A78<AN%
%
𝑖ℏ
𝜕Ψ(𝑟,t)
𝜕𝑡 =−
ℏ!
2𝑚𝛥Ψ𝑟,t+Vr,𝑡.Ψ𝑟,t%
%
"-AA-%,T7<A?:B%-;A%8?B,<?>-J%%-B%<++:>=%<P-+%8-%@>?B+?@-%=-%;7@->@:;?A?:BN%
%
"-AA-%,T7<A?:B%B-%=:?A%@<;%iA>-%C,C:>?;,-N%
%
eB%B-%+:B;?=S>-%=<B;%8<%;7?A-%=7%+:7>;%T7-%=-;%F:B+A?:B;%=D:B=-%<%7B-%=?C-B;?:B%R%%
Ψ𝑟,t=!Ψ(x,t)%
)D,T7<A?:B%=-%.+9>u=?BU->%;D,+>?A%<8:>;%R%%
%
𝑖ℏ
𝜕Ψ(𝑥,t)
𝜕𝑡 =−
ℏ!
2𝑚
𝜕!Ψ𝑥,t
𝜕𝑥!+Vx,𝑡.Ψ𝑥,t%
%
F8 .#'&%<BC()C%E'()+))
%
v,F?B?A?:B%R%7B-%@<>A?+78-%8?O>-%-;A%7B-%@<>A?+78-%T7?%BD-;A%;:7C?;-%E%<7+7B-%?BA-><+A?:B%l%:B%@-7A%
+:B;?=,>->%T7-%R%
QIGJAK%V%aN%
%
v<B;%8<%;7?A-%=-%+-%+9<@?A>-%:B%B-%+:B;?=S>-%T7-%=-;%@<>A?+78-;%8?O>-;N%
%
)D,T7<A?:B%=-%.+9>u=?BU->%;D,+>?A%<8:>;%R%%
%
𝑖ℏ
𝜕Ψ(𝑥,t)
𝜕𝑡 =−
ℏ!
2𝑚
𝜕!Ψ𝑥,t
𝜕𝑥!%
%
%
H8 !#?)=(?)@&#&?)?&#&%:;;#%'(?)P):;=(?)=()=()7':OC%()+)
%
3@Q%;%&%:;%R%:B%<@@-88-%,A<A;%;A<A?:BB<?>-;%=-;%,A<A;%@:7>%8-;T7-8;%8<%F:B+A?:B%=D:B=-%@-7A%;D,+>?>-%R%%
Ψ𝑥,t=𝜑𝑥.𝑓(𝑡)%
rIGK%-A%FIAK%,A<BA%E%@>?:>?%=-;%F:B+A?:B;%E%P<8-7>;%<:D$C(R(?N%
%
eB%C:BA>-%<8:>;%T7-%R%%%
𝑓𝑡=𝑒!!"# %
%
)D,B->U?-%=-%+-A%,A<A%;A<A?:BB<?>-%-;A%R%
1%V%ℏ.𝜔%
%
!:7>%7B%,A<A%;A<A?:BB<?>-%+-AA-%,B->U?-%-;A%F?G,-N%
%
)<%=-B;?A,%=-%@>:O<O?8?A,%-;A%<8:>;%R%%
Ψ𝑥,t!=𝜑𝑥.𝑓(𝑡)!=𝜑𝑥!%
%

!"#!"$%&'#&(% % )*+,-%."/0123415%6789:7;-%
%
)<%=-B;?A,%=-%@>:O<O?8?A,%=D7B%,A<A%;A<A?:BB<?>-%-;A%?B=,@-B=<BA-%=7%A-C@;N%
%
)D,T7<A?:B%P,>?F?,-%@<>%8<%@<>A?-%;@<A?<8-%=-%8<%F:B+A?:B%=D:B=-%-;A%R%%
%
−
ℏ!
2𝑚𝛥φx=𝐸.𝜑(𝑥)%
=:BA%8<%;:87A?:B%-;A%R%%
φx=𝐴.𝑒!!"# +𝐵.𝑒!"# %
<P-+%R%
!𝑘=!
2𝑚𝐸
ℏ!=
2𝑚𝜔
ℏ%
%
)<%F:B+A?:B%=D:B=-%=D7B%,A<A%;A<A?:BB<?>-%=D7B-%@<>A?+78-%8?O>-%;D,+>?A%=:B+%R%%
Ψ𝑥,t=𝐴.𝑒!!(!"!!")+𝐵.𝑒!!(!"!!")=𝐴.𝑒!!(!"!!")/ℏ+𝐵.𝑒!!(!"!!")/ℏ%
%
eB%>-C<>T7-%T7-%+-AA-%F:B+A?:B%=D:B=-%-;A%<B<8:U7-%E%7B-%:B=-%@8<B-%,8-+A>:C<UB,A?T7-N%
%
5-8<A?:B%=-%=?;@->;?:B%R%%
𝜔=!
ℏ𝑘!
2𝑚%
Q?A-;;-%=-%@9<;-%R%%
𝑣!=
𝜔
𝑘=
ℏ𝑘
2𝑚%
%
)<%@>:@<U<A?:B%=D7B-%@<>A?+78-%8?O>-%-;A%=?;@->;?P-N%
%
S8 .#GB(&)=>:;=(?)P)'(C#&%:;)=>%;<('&%&B=(8)
%
)D:B=-%C:B:+9>:C<A?T7-%>-@>,;-BA-%C<8%8-;%@<>A?+78-;%T7<BA?T7-;J%+<>%8<%+:B=?A?:B%=-%B:>C<8?;<A?:B%
=?P->U-N%eB%=:?A%+:B;?=,>->%7B%@<T7-A%=D:B=-;%+:B;A?A7,%=D:B=-;%,8,C-BA<?>-;%=-%8<%F:>C-%R%%
Ψ𝑥,t=𝐴!.𝑒!!(!!!!!!!)%
%
1B%U,B,><8%:B%+:B;?=S>-%7B%@<T7-A%=D:B=-;%F:>C,%=D7B%-B;-CO8-%+:BA?B7%=-%P-+A-7>;%=D:B=-;%-A%=-%
@78;<A?:B;%<;;:+?,-;J%P:?;?B;%=D7B%P-+A-7>%=D:B=-%+-BA><8%wa%=D7B-%@78;<A?:B%+-BA><8-%xa%+:>>-;@:B=<BA%E%
waJ%-A%+:C@>?;-;%=<B;%7B%?BA->P<88-%yw&J%wmzN%
v<B;%7B%?BA->P<88-%=w%<7A:7>%=D7B-%P<8-7>%wJ%8D<C@8?A7=-%-;A%UIwKN=wJ%-A%8-%@<T7-A%=D:B=-%@-7A%;D,+>?>-%R%%
Ψ𝑥,t=𝑔𝑘.𝑒!!(!"!!(!)!)𝑑𝑘
!!
!!
%
%
)<%P?A-;;-%=-%U>:7@-%-;A%R%%
%
𝑣
!=
𝑑𝜔
𝑑𝑘 =
ℏ𝑘
𝑚=𝑝 𝑚=𝑣%
%
)<%P?A-;;-%=-%U>:7@-%=7%@<T7-A%=D:B=-%;D?=-BA?F?-%E%8<%P?A-;;-%=-%8<%@<>A?+78-N%
%
/#'O(B')?$#&%#C()=B)$#GB(&)=>:;=()(;);:DE'()=>:;=()T)!1)U)+))
%
":B;?=,>:B;%7B%@<T7-A%=:BA%8<%F:B+A?:B%UIwK%<%8D<887>-%+?W+:BA>-%R%%
%
)<%8<>U-7>%=7%@<T7-A%=D:B=-;%-;A%sw%I+D-;A%@<>%-G-C@8-%8<%8<>U-7>%E%
C?W9<7A-7>KN%
sw%
%
%
wa%
w%
UIwK%

!"#!"$%&'#&(% % )*+,-%."/0123415%6789:7;-%
%
%
.7@@:;:B;%8-%@<T7-A%+:B;A?A7,%=-%8<%;:CC-%=?;+>SA-%=-%A>:?;%:B=-;%@8<B-;%=-%P-+A-7>;%waJ%waWsw#m%-A%%
waYsw#m%=D<C@8?A7=-;%>-;@-+A?P-;%qJ%q#m%-A%q#mN%
5-C<>T7-%R%q%-;A%@>:@:>A?:BB-88-%E%UIwaKN%
%
q%8D?B;A<BA%AVaJ%:B%<%R%%
Ψ𝑥,0=𝐴.𝑒!!!!+
1
2𝑒!(!!!∆!
!)!)+
1
2𝑒!(!!!∆!
!)!)%
Ψ𝑥,0=𝐴.𝑒!!!!1+cos!(
∆k
2x)%
eB%+:B;A<A-%T7-% Ψ𝑥,0%-;A%C<G?C<8-%@:7>%G%V%a%l%8-;%A>:?;%:B=-;%?BA->FS>-BA%<8:>;%+:B;A>7+A?P-C-BAN%
%
%
)D?BA->F,>-B+-%-;A%=-;A>7+A?P-%@:7>%∆!
!
x=π%:7%∆!
!
x=−π%l%8<%8<>U-7>%=7%@<T7-A%=D:B=-;%-;A%=:B+%R%%
∆x=
4π
∆k%
⟺∆k.∆x=4π%
%
eB%+:B;A<A-%T7-%8D-GA-B;?:B%;@<A?<8-%=7%@<T7-A%=D:B=-;%-;A%=D<7A<BA%@87;%U><B=-%T7-%;:B%-GA-B;?:B%-B%w%
-;A%@-A?A-J%+:BF:>C,C-BA%E%8D?B,U<8?A,%=-%/-?;-BO->U%R%%
∆k.∆x≧1/2%
%
5-C<>T7-%R%8<%F:>C-%=7%@<T7-A%=D:B=-;%=:BB,%-;A%@,>?:=?T7-%-B%GJ%-A%@>,;-BA-%=:B+%7B-%;,>?-%=-%
C<G?C<;%-A%=-%C?C?C<;N%"-+?%@>:P?-BA%=7%F<?A%T7-%HIGJaK%-;A%8<%;7@->@:;?A?:B%=D7B%B:CO>-%F?B?%=D:B=-;%l%
@:7>%7B-%;7@->@:;?A?:B%+:BA?B7-J%+-+?%B-%;-%@>:=7?A%@<;%-A%HIGJaK%B-%@-7A%<P:?>%T7D7B%;-78%C<G?C7CN%