MathematicalTools

advertisement
Mathematical tools for QM
1. Complex Exponential
A basic understanding of complex calculus will help enormously with grasping the ins and outs of Quantum Mechanics. In particular, the function eix
plays a pivotal role in all of this. Why? Because functions of this form are
general solutions of the wave equation, and Quantum Mechanics happens to be
based on the Schrödinger’s Wave Equation. The complex exponential is defined
as follows:
eix = cos(x) + i · sin(x)
(1)
with i, of course, defined as i2 = −1. From (1) the wave-character of the
complex
exponential is clear. In fact, it consists
of two waves, a Real part
Re eix = cos(x), and an Imaginary part Im eix = sin(x). Importantly,
these waves are always π/2 ‘out-of-phase’ with each other (remember that
cos(x) = sin(x + π/2)). The Real and Imaginary parts are therefore said to
be orthogonal to each other. The complex exponential is often thought of as a
‘phase function’, with an amplitude of 1 and a phase x as defined by the angle
between the Real and Imaginary parts. Below are some useful expressions related to this.
Square modulus
ix 2
e = eix · e−ix
=
{cos(x) + i · sin(x)} · {cos(x) − i · sin(x)}
=
cos2 (x) + sin2 (x)
=
1
(2)
Amplitude
ix q
e = |eix |2 = 1
Phase
Θ ≡ arctan
!
Im eix
sin(x)
= arctan
=x
Re {eix }
cos(x)
(3)
(4)
Special phase settings (with n = ± 0, 1, 2, 3....)
e
e
ei2nπ
=
1
i(2n+1)π
=
−1
i(2n+1) π
2
=
4
(5)
(6)
n
(−1) i
(7)
2. Useful Trigonometric Identities
sin2 (x) + cos2 (x)
=
1
(8)
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos2 (x) − sin2 (x)
=
cos2 (x)
=
sin2 (x)
=
sin3 (x)
=
cos3 (x)
=
sin(x)
=
cos(x)
=
1 − 2 sin2 (x)
1
{1 + cos(2x)}
2
1
{1 − cos(2x)}
2
1
{3 sin(x) − sin(3x)}
4
1
{3 cos(x) + cos(3x)}
4
eix − e−ix
2i
eix + e−ix
2
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
3. Table of Useful Integrals
where a > 0 and n is an integer
2π
Z
sin2 (nx)dx
= π
(17)
sin(ax2 )dx
=
(18)
cos(ax2 )dx
Z ∞
e−ax dx
=
0
∞
Z
0
Z ∞
0
=
0
∞
Z
xn e−ax dx
= n!/an+1
r
1 π
=
2 a
1
=
2a
r
1
π
=
4a a
0
Z
Z
Z
∞
0
∞
2
e−ax dx
0
∞
{n 6= 0}
r
1 π
2 2a
r
1 π
2 2a
1
a
2
xe−ax dx
2
x2 e−ax dx
0
5
(19)
(20)
(21)
(22)
(23)
(24)
4. Rules for Differentiation and Integration
d
(f g)
dx
dg
df
+g
dx
dx
df {g(x)} dg(x)
d
(f {g(x)}) =
dx
dg(x)
dx
Z Z dg
df
f
dx = f g − g
dx
dx
dx
= f
6
(25)
(26)
(27)
Download
Related flashcards
Create Flashcards