Tatsu Takeuchi
Virginia Tech
@Osaka University, January 10, 2012
Work in collaboration with:
 Zack Lewis
 Djordje Minic
 Lay Nam Chang
Also contributions from:
 Naotoshi Okamura (Yamanashi Univ.)
 Sandor Benczik (quit physics)
 Saif Rayyan (MIT, physics education)
The Minimal Length Uncertainty Relation

h  1
x     p  x  xmin  h 
2 p


Suggested by Quantum Gravity. Observed in perturbative String Theory.
Deformed Commutation Relation
1
xˆ , pˆ   1  pˆ 2

ih


h  1
 x     p
2 p

The operators can be represented as :

2 d
ˆ
ˆ
x

ih
1

p



dp

 pˆ  p

and the inner product as

dp
*
f g  
f
(p) g(p)
2
 (1 p )
Harmonic Oscillator
2
ˆ
p
1 2
ˆ
H
 k xˆ
2m 2
2
 2 
2 

p
h
k
d
 (p)  E (p)
(1  p 2 )  
 
dp  2m 

 2 

Can be solved exactly. Energy eigenvalues

:

 2
k  1 
1 
4
4
2
En  n   (xmin )  4a  n  n  ( xmin ) 

2  2 
2 

2
h
h
4
a

km
m
No longer evenly spaced. n2 - dependence is introduced.
Multi Dimensional Case
 
pˆ , pˆ  0
1
xˆ , xˆ  (2    )   (2    ) pˆ Lˆ ,

ih
1
xˆ i , pˆ j  1  pˆ 2 ij   pˆ i pˆ j
ih
i
j
2
i
j
ij
ˆ ˆ ˆ ˆ
ˆL  xi pj  xj pi
ij
1  pˆ 2
The operators can be represented as :



 
D 1


2

 xˆ i  ih1  pˆ    pi pj
    
  (    ) pi 
 2 
pi
pj 

 




 pˆ i  pi
and the inner product as

dp
*
f g  
f
(p) g(p)
2 
 [1 (    )p ]
Isotropic Harmonic Oscillator in D dimensions
D (p1, p2,L , pD )  R(p)Yl m D 2m D 3 L m 2m1 ()
2

2
2

(D
1)(1

p
)[1
(



)
p
] 
h k 

2
[1 (    ) p ]  

2 
p 
p
p

L2 (1  p 2 ) 2 
p2

R(p)  E R(p),
L2  l (l  D  2)
R(p) 
2
p
2m

2

Can be solved exactly. Energy eigenvalues

:
k  D 
4
Enl  n   xmin
4L2  (D  ) 2  4a 4

2  2 
2

 2 D 2  D 


D

2


 xmin (1 ) n    (1  ) L 


,
x

h

,


.

min
 2 
4  2 




Dependence on angular momentum introduced. SU(D) degeneracy is broken.
2D Case
Some Details (1D case) :
Change variable to :
1

arctan  p


d
ˆ
x  ih


d

 pˆ  1 tan 





2 


2 
 /2 
f g 
 d f
*
( ) g( )
 / 2 
Schrodinger Equation :
 h2 k d 2

1
2
tan   ( )  E  ()
2 
 2 d 2m

Infinite square - well problem for large n, and also in the limit m .
Solution:
The solution is :

( )
n



n!(n


)

4
( )   2 (  )
 cos  C n sin 
2 (n  2  ) 


 
where :
4


1
1
a
 
 

2
4 xmin 
1  
Uncertainties :
x  xmin
(   n)[(2  1)n  
,
(2  1)
p 
1

2n 1
2  1

Wave-functions:
n=1
n=0
n=2
Uncertainties of the Harmonic Oscillator: β=0
Uncertainties of the Harmonic Oscillator: β≠0
How can we get onto the Δx~Δp branch?
Harmonic Oscillator with negative mass:
2
ˆ
p
1
ˆ 
H
 k xˆ 2
2m 2
2
 2 
2 

p
h
k
d
 (p)  E (p)
(1  p 2 )  
 
dp  2 m 

 2 

Energy eigenvalues :


 2
k  1 
1 
4
4
2
En  n   (xmin )  4a  n  n  ( xmin ) 

2  2 
2 

a4
h2
km
Harmonic Oscillator with negative mass:
The solution is the same as before


n!(n  ) 

 ( )   2 ()
 cos  C n sin 
2 (n  2 ) 

except :
( )
n
4
4


1
1
a
 
 

2
4 xmin 


 
1

    1, xmin  2a
2


Uncertainties of the Harmonic Oscillator: β≠0, m<0
Classical Limit:
1
xˆ, pˆ   1  pˆ 2  

ih
x, p 1  p 
2
Classical Equations of Motion :

pÝ p, H
xÝ x, H,
Liouville Theorem :
dx  dp

dx  dp
1  p 2
h1  p 2  can be considered a p - dependent effective h(p).
Classical Harmonic Oscillator:
Hamiltonian :
p2 1 2
H
 kx
2m 2
Classical Equations of Motion :

1
2
Ý
x

x,
H

1

p
p





m

pÝ p, H k 1  p 2 x

Time - dependence of x and p are different, but the trajectories in
phase space are the same as the  = 0 case since the Hamiltonian
is the same.
m>0 case:
m limit:
m<0 case:
Compactification:
Classical Probabilities:
Comparison with Quantum Probabilities:
Work in progress:
Other potentials :
V(x)  F x
F x
V(x)  

(x  0)
(x  0)
 Discreate energy eigenstates have been found for the
negative mass case.
 Uncertainties are difficult to calculate. What do we
mean by x=0?
Conclusions:
 The minimal length uncertainty relation allows
discreate energy “bound” states for “inverted”
potentials.
 In the classical limit, these “bound” states can be
understood to be due to the finite time the particle
spends near the phase-space origin.
 Particles move at arbitrary large velocites. Do the nonrelativistic negative mass states correspond to
relativistic imaginary mass tachyons?