Question

A wave function is given by
ψ(x) = 0 if, < 0
= (A√30)x(L − ) if, 0 ≤ ≤ L
= 0 if, > L
what is value of A? ​

Answer 1

Answer:

angular frequency ω and group velocity vg for a nonrelativistic particle of mass m are:

p = mv =

h

λ

= ~k (7)

E =

1

2

mv2 =

p

2

2m

=

~

2k

2

2m

= ~ω (8)

vg =

dω

dk = v (9)

When k = 50 nm−1

,

λ = 126 pm p = 9.87 keV/c (10)

and, for an electron (m = 511 keV/c2

),

E = 95.2 eV v = 1.93 × 10−2

c (11)

The equations relating the speed v, momentum p, de Broglie wavelength λ, wave number k, total energy

E, kinetic energy K, angular frequency ω and group velocity vg for a relativistic particle of mass m are:

p = γmv =

h

λ

= ~k (12)

E = γmc2 = mc2 + K =

p

p

2c

2 + m2c

4 = ~ω (13)

vg =

dω

dk = v =

pc2

E

(14)

γ =

1

p

1 − β

2

(15)

β = v/c (16)

When k = 50 pm−1

,

λ = 126 fm p = 9.87 MeV/c (17)

and, for an electron (m = 511 keV/c2

),

E = 9.88 MeV K = 9.37 MeV v = 0.9987c (18)

Potential Energy of a Particle

Problem 5.5, page 224

In a region of space, a particle with mass m and with zero energy has a time-independent wave function

ψ(x) = Axe−x

2/L2

(19)

where A and L are constants.

•Determine the potential energy U(x) of the particle.

Solution