Question

True or False:

a.) Probability density can never be negative.

b.) The state f(x), a wave function can never be real (f(x)).

c.)The state f(x), a wave function must be real number.

d.) If z=z*, then z must be a real number.

e.) integral from neg infinity to infinity of the wave function equals 1 for a real particle in a 1d system.

f.) The product of a number and its complex conjugate is always a real number.

Answer 1

a.) TRUE

Probability density can never be Because the probability density function is the derivative of the distribution function and we know that distribution function is an increasing function on R thus its derivative ( i.e. density function ) will be always positive.

b.) TRUE

The state f(x), a wave function can never be real (f(x)). It is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. it's modulus square is observable and thus should be real (it gives probability density).

c.) FALSE

from above ,it is clear that The state f(x), a wave function may not be real number.

d.)TRUE

If z=z*, then z must be a real number. because only real number can be equal to real quantity and imaginary quantity may be equal to imaginary number .it is not possible to equate real quantity to imaginary quantity .

In other way we can understand it as following types

let z=x+iy

z=x+iy where, x,y∈R

First assume that we are given z is real then its imaginary part must be equal to 0
Therefore, z=x

Now, assume that z=z*

where z*=x−iy

then,

x+iy=x−iy

iy=−iy

2y=0

that means y=o
So,we get, z=x

e.) TRUE

integral from neg infinity to infinity of the wave function equals 1 for a real particle in a 1d system because we are 100% sure that particle will lie in this region anywhere.so probability = 1

f.) TRUE

The product of a number and its complex conjugate is always a real number.

LET a number z= x+iy

Then its complex conjugate will be z*= x-iy

Now zz*= (x+iy)(x-iy)

zz*= x² +y²  

Here zz* is a real number .

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