Matter Waves and de Broglie Relation

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A particle of mass $M$ at rest decays into two particles of masses $m_{1}$ and $m_{2}$ having non-zero velocities. The ratio of the de-Broglie wavelengths of the particles $\frac{λ _{1}}{λ _{2}}$ is

\frac{m _{1}}{m _{2}}

(B)

\frac{m _{2}}{m _{1}}

(C)

1

(D)

\frac{m _{2}}{m _{1}}

JEE Advanced 1997

LEVELJEE Advanced

Assume that the de-Broglie wave associated with an electron can form a standing wave between the atoms arranged in a one dimensional array with nodes at each of the atomic sites. It is found that one such standing wave is formed if the distance $d$ between the atoms of the array is $2 \overset{˚}{A}$ . A similar standing wave is again formed if $d$ is increased to $2.5 \overset{˚}{A}$ but not for any intermediate value of $d$ . Find the energy of the electron in eV and the least value of $d$ for which the standing wave of the type described above can form.

LEVELJEE Main

The energy of a photon is equal to the kinetic energy of a proton. The energy of the photon is $E$ . Let $λ_{1}$ be the de-Broglie wavelength of the proton and $λ_{2}$ be the wavelength of the photon. The ratio $\frac{λ _{1}}{λ _{2}}$ is proportional to

(A)

E^{0}

(B)

E^{1/2}

(C)

E^{- 1}

(D)

E^{- 2}

JEE Advanced 2009

LEVELJEE Advanced

Comprehension Passage

When a particle is restricted to move along

x

-axis between

x = 0

and

x = a

, where

a

is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends

x = 0

and

x = a

. The wavelength of this standing wave is related to the linear momentum

p

of the particle according to the de-Broglie relation. The energy of the particle of mass

m

is related to its linear momentum as

E = \frac{p ^{2}}{2 m}

. Thus, the energy of the particle can be denoted by a quantum number

n

taking values

1, 2, 3, \dots (n = 1

, called the ground state) corresponding to the number of loops in the standing wave. Use the model described above to answer the following three questions for a particle moving in the line

x = 0

x = a

. [Take

h = 6.6 \times 1 0^{- 34}

Js and

e = 1.6 \times 1 0^{- 19}

Question 1:

The allowed energy for the particle for a particular value of $n$ is proportional to

(A)

a^{- 2}

(B)

a^{- 3/2}

(C)

a^{- 1}

(D)

a^{2}

Question 2:

If the mass of the particle is $m = 1.0 \times 1 0^{- 30}$ kg and $a = 6.6$ nm, the energy of the particle in its ground state is closest to

(A)

0.8

meV

(B)

8

meV

(C)

80

meV

(D)

800

meV

Question 3:

The speed of the particle that can take discrete values is proportional to

(A)

n^{- 3/2}

(B)

n^{- 1}

(C)

n^{1/2}

(D)

n

JEE Advanced 2010

LEVELJEE Main

An $α$ -particle and a proton are accelerated from rest by a potential difference of $100 V$ . After this, their de-Broglie wavelengths are $λ_{α}$ and $λ_{p}$ respectively. The ratio $\frac{λ _{p}}{λ _{α}}$ , to the nearest integer, is

JEE Advanced 2013

LEVELJEE Advanced

A proton is fired from very far away towards a nucleus with charge $Q = 120 e$ , where $e$ is the electronic charge. It makes a closest approach of $10 fm$ to the nucleus. The de-Broglie wavelength (in units of fm) of the proton at its start is [Take the proton mass, $m_{p} = (5/3) \times 1 0^{- 27} kg$ ; $h = 6.63 \times 1 0^{- 34} J-s$ ; $e = 1.6 \times 1 0^{- 19} C$ ; $\frac{1}{4 π ε _{0}} = 9 \times 1 0^{9} m/F$ ; $1 fm = 1 0^{- 15} m$ ]

JEE Advanced 2007

LEVELJEE Main

Electrons with de-Broglie wavelength $λ$ fall on the target in an X-ray tube. The cut-off wavelength of the emitted X-rays is

(A)

λ_{0} = \frac{2 m c λ ^{2}}{h}

(B)

λ_{0} = \frac{2 h}{m c}

(C)

λ_{0} = \frac{2 m ^{2} c ^{2} λ ^{3}}{h ^{2}}

(D)

λ_{0} = λ

JEE Advanced 2005

LEVELJEE Main

A beam of electron is used in an YDSE experiment. The slit width is $d$ . When the velocity of electron is increased, then

(A)

no interference is observed

(B)

fringe width increases

(C)

fringe width decreases

(D)

fringe width remains same

JEE Advanced 2005

LEVELJEE Main

The potential energy of a particle varies as $U (x) = E_{0}$ for $0 \leq x \leq 1$ $= 0$ for $x > 1$ For $0 \leq x \leq 1$ , de-Broglie wavelength is $λ_{1}$ and for $x > 1$ the de-Broglie wavelength is $λ_{2}$ . Total energy of the particle is $2 E_{0}$ . Find $\frac{λ _{1}}{λ _{2}}$ .

JEE Advanced 2016

LEVELJEE Advanced

Light of wavelength $λ_{ph}$ falls on a cathode plate inside a vacuum tube as shown in the figure. The work function of the cathode surface is $ϕ$ and the anode is a wire mesh of conducting material kept at a distance $d$ from the cathode. A potential difference $V$ is maintained between the electrodes. If the minimum de Broglie wavelength of the electrons passing through the anode is $λ_{e}$ , which of the following statements(s) is (are) true?

* Multiple Correct Options

(A)

λ_{e}

increases at the same rate as

λ_{ph}

for

λ_{ph} < \frac{h c}{ϕ}

(B)

λ_{e}

is approximately halved, if

d

is doubled

(C)

λ_{e}

decreases with increase in

ϕ

and

λ_{ph}

(D)

For large potential difference (

V >> \frac{ϕ}{e}

λ_{e}

is approximately halved if

V

is made four times.

JEE Main 2021

LEVELJEE Advanced

Consider two separate ideal gases of electrons and protons having same number of particles. The temperature of both the gases are same. The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to

(A)

(\frac{m _{p}}{m _{e}})^{\frac{3}{2}}

(B)

\frac{m _{e}}{m _{p}}

(C)

\frac{m _{p}}{m _{e}}

(D)

\frac{m _{p}}{m _{e}}

JEE Main 2021

LEVELJEE Main

A particle is travelling $4$ times as fast as an electron. Assuming the ratio of de-Broglie wavelength of a particle to that of electron is $2 : 1$ , the mass of the particle is

(A)

\frac{1}{16}

times the mass of electron

(B)

8

times the mass of electron

(C)

16

times the mass of electron

(D)

\frac{1}{8}

times the mass of electron

JEE Main 2020

LEVELJEE Main

An electron (mass $m$ ) with initial velocity $v = v_{0} \hat{i} + v_{0} \hat{j}$ is in an electric field $E = - E_{0} \hat{k}$ . If $λ_{0}$ is initial de-Broglie wavelength of electron, then its de Broglie wavelength at time $t$ is given by

(A)

\frac{λ _{0}}{1 + \frac{e ^{2} E _{0}^{2} t ^{2}}{m ^{2} v _{0}^{2}}}

(B)

\frac{λ _{0}}{2 + \frac{e ^{2} E _{0}^{2} t ^{2}}{m ^{2} v _{0}^{2}}}

(C)

\frac{λ _{0}}{1 + \frac{e ^{2} E _{0}^{2} t ^{2}}{2 m ^{2} v _{0}^{2}}}

(D)

\frac{λ _{0}}{2 + \frac{e ^{2} E _{0}^{2} t ^{2}}{2 m ^{2} v _{0}^{2}}}

JEE Main 2015

LEVELBoard

Match List I (fundamental experiment) with List II (its conclusion) and select the correct option from the choices given below the list. $List I A. Franck-Hertz experiment B. Photo-electric experiment C. Davisson-Germer experiment List II 1. Particle nature of light 2. Discrete energy levels of atom 3. Wave nature of electron 4. Structure of atom$

(A)

\to

1, B

\to

4, C

\to

(B)

\to

2, B

\to

4, C

\to

(C)

\to

2, B

\to

1, C

\to

(D)

\to

4, B

\to

3, C

\to

JEE Main 2021

LEVELJEE Main

The de-Broglie wavelength of a proton and $α$ -particle are equal. The ratio of their velocities is

A particle A of mass $m$ and charge $q$ is accelerated by a potential difference of $50 V$ . Another particle B of mass $4 m$ and charge $q$ is accelerated by a potential difference of $2500 V$ . The ratio of de-Broglie wavelengths $\frac{λ _{A}}{λ _{B}}$ is close to

(A)

4.47

(B)

10.00

(C)

0.07

(D)

14.14

JEE Main 2020

LEVELJEE Main

An electron of mass $m$ and magnitude of charge $∣ e ∣$ initially at rest gets accelerated by a constant electric field $E$ . The rate of change of de Broglie wavelength of this electron at time $t$ ignoring relativistic effects is

(A)

\frac{h}{∣ e ∣ E t}

(B)

- \frac{h}{∣ e ∣ E t}

(C)

\frac{- h}{∣ e ∣ E t ^{2}}

(D)

\frac{∣ e ∣ E t}{h}

JEE Main 2020

LEVELJEE Main

A particle is moving 5 times as fast as an electron. The ratio of the de-Broglie wavelength of the particle to that of the electron is $1.878 \times 1 0^{- 4}$ . The mass of the particle is close to

(A)

4.8 \times 1 0^{- 27}

(B)

9.1 \times 1 0^{- 31}

(C)

1.2 \times 1 0^{- 28}

(D)

9.7 \times 1 0^{- 28}

JEE Main 2021, 25 July Shift-II

LEVELJEE Main

An electron moving with speed $v$ and a photon moving with speed $c$ , have same de-Broglie wavelength. The ratio of kinetic energy of electron to that of photon is

(A)

\frac{3 c}{v}

(B)

\frac{v}{3 c}

(C)

\frac{v}{2 c}

(D)

\frac{2 c}{v}

JEE Main 2021

LEVELBoard

An electron of mass $m_{e}$ and a proton of mass $m_{p} = 1836 m_{e}$ are moving with the same speed. The ratio of their de-Broglie wavelength $\frac{λ _{electron}}{λ _{proton}}$ will be

\frac{1}{1836}

Two particles move at right angle to each other. Their de-Broglie wavelengths are $λ_{1}$ and $λ_{2}$ , respectively. The particles suffer perfectly inelastic collision. The de-Broglie wavelength $λ$ of the final particle, is given by

(A)

\frac{1}{λ ^{2}} = \frac{1}{λ _{1}^{2}} + \frac{1}{λ _{2}^{2}}

(B)

λ = λ_{1} λ_{2}

(C)

λ = \frac{λ _{1} + λ _{2}}{2}

(D)

\frac{2}{λ} = \frac{1}{λ _{1}} + \frac{1}{λ _{2}}

LEVELJEE Main

In an experiment, electrons are made to pass through a narrow slit of width $d$ comparable to their de-Broglie wavelength. They are detected on a screen at a distance $D$ from the slit (see figure). Which of the following graphs can be expected to represent the number of electrons $N$ detected as a function of the detector position $y$ ( $y = 0$ corresponds to the middle of the slit) ?

A beam of electrons of energy $E$ scatters from a target having atomic spacing of $1 \overset{˚}{A}$ . The first maximum intensity occurs at $θ = 6 0^{\circ}$ . Then, $E$ (in eV) is ......... . (Given, Planck's constant, $h = 6.64 \times 1 0^{- 34} Js$ , $1 eV = 1.6 \times 1 0^{- 19} J$ and electron mass, $m = 9.1 \times 1 0^{- 31} kg$ .)

LEVELJEE Advanced

Comprehension Passage

Wave property of electrons implies that they will show diffraction effects. Davisson and Germer demonstrated this by diffracting electrons from crystals. The law governing the diffraction from a crystal is obtained by requiring that electron waves reflected from the planes of atoms in a crystal interfere constructively (see figure).

Question 1:

Electrons accelerated by potential $V$ are diffracted from a crystal. If $d = 1 \overset{˚}{A}$ and $i = 3 0^{\circ}$ , $V$ should be about ( $h = 6.6 \times 1 0^{- 34} J-s$ , $m_{e} = 9.1 \times 1 0^{- 31} kg$ , $e = 1.6 \times 1 0^{- 19} C$ )

An $α$ -particle and a proton are accelerated from rest by a potential difference of $200 V$ . After this, their de-Broglie wavelengths are $λ_{α}$ and $λ_{p}$ , respectively. The ratio $\frac{λ _{p}}{λ _{α}}$ is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The allowed energy for the particle for a particular value of n is proportional to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the mass of the particle is m=1.0×10−30 kg and a=6.6 nm, the energy of the particle in its ground state is closest to

The speed of the particle that can take discrete values is proportional to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Electrons with de-Broglie wavelength λ fall on the target in an X-ray tube. The cut-off wavelength of the emitted X-rays is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A beam of electron is used in an YDSE experiment. The slit width is d. When the velocity of electron is increased, then

Consider two separate ideal gases of electrons and protons having same number of particles. The temperature of both the gases are same. The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A particle is travelling 4 times as fast as an electron. Assuming the ratio of de-Broglie wavelength of a particle to that of electron is 2:1, the mass of the particle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The de-Broglie wavelength of a proton and α-particle are equal. The ratio of their velocities is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } An electron moving with speed v and a photon moving with speed c, have same de-Broglie wavelength. The ratio of kinetic energy of electron to that of photon is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } An electron of mass me​ and a proton of mass mp​=1836me​ are moving with the same speed. The ratio of their de-Broglie wavelength λproton​λelectron​​ will be

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Electrons accelerated by potential V are diffracted from a crystal. If d=1 A˚ and i=30∘, V should be about (h=6.6×10−34 J-s, me​=9.1×10−31 kg, e=1.6×10−19 C)

The allowed energy for the particle for a particular value of $n$ is proportional to

If the mass of the particle is $m = 1.0 \times 1 0^{- 30}$ kg and $a = 6.6$ nm, the energy of the particle in its ground state is closest to

Electrons with de-Broglie wavelength $λ$ fall on the target in an X-ray tube. The cut-off wavelength of the emitted X-rays is

A beam of electron is used in an YDSE experiment. The slit width is $d$ . When the velocity of electron is increased, then

A particle is travelling $4$ times as fast as an electron. Assuming the ratio of de-Broglie wavelength of a particle to that of electron is $2 : 1$ , the mass of the particle is

The de-Broglie wavelength of a proton and $α$ -particle are equal. The ratio of their velocities is

An electron moving with speed $v$ and a photon moving with speed $c$ , have same de-Broglie wavelength. The ratio of kinetic energy of electron to that of photon is

An electron of mass $m_{e}$ and a proton of mass $m_{p} = 1836 m_{e}$ are moving with the same speed. The ratio of their de-Broglie wavelength $\frac{λ _{electron}}{λ _{proton}}$ will be

Electrons accelerated by potential $V$ are diffracted from a crystal. If $d = 1 \overset{˚}{A}$ and $i = 3 0^{\circ}$ , $V$ should be about ( $h = 6.6 \times 1 0^{- 34} J-s$ , $m_{e} = 9.1 \times 1 0^{- 31} kg$ , $e = 1.6 \times 1 0^{- 19} C$ )